f0945ea a7d41c1
================================================================================
A TORSION-FREE GROUP ALGEBRA THAT IS NOT DIRECTLY FINITE
A plain-English companion version, written for curious high school readers
================================================================================
PURPOSE OF THIS DOCUMENT
------------------------
The original paper (torsion_free_group_not_directly_finite_original.txt,
about 119 KB) is a research paper in contemporary group theory and
non-commutative algebra. It answers a question of Irving Kaplansky from
1972 with a NO: there is a group algebra over the two-element field that
is not "directly finite", and the group can be chosen finite-presented
and torsion-free. The audience of that paper is other group theorists
and ring theorists.
This companion version is written for a different audience: a reader
who knows high-school algebra (variables, matrices, the idea of a
proof) and is willing to think through a construction step by step,
but who has not seen university algebra. It is meant to be read
alongside the original. Where the original proves a technical lemma,
this version says WHAT the lemma says in plain words, WHY it is true,
and WHY the whole construction needs it.
The document has these parts:
PURPOSE OF THIS DOCUMENT (this section)
PART I -- the question, the objects (groups, fields, group rings),
and the shape of the answer, in words.
PART II -- exact statements of the main theorem and every major
intermediate result, restated cleanly.
PART III -- the whole proof, as twelve short steps, no details.
PART IV -- a glossary of every specialized word the paper uses.
PART V -- the abstract and the conventions of the paper.
PART VI -- small concrete calculations a reader can do by hand.
PART VII -- why this matters, and what stays open.
PART VIII -- the lemma and result index of the original paper.
PART IX -- exact statements of the results, verbatim from the
original, lightly retypeset for ASCII.
"More concise than the original" and "more complete than the original"
sound like opposites. They are not. The original is long because it
has to justify every step to a professional who will check it. This
companion is short because it only needs to explain the shape of the
argument to a reader who is not going to referee it. No part of the
original's mathematical content is silently dropped; some things are
compressed, and things the original takes for granted are spelled out.
PART I. WHAT THIS PAPER IS ABOUT
================================================================================
1. THE QUESTION IN ONE SENTENCE
-------------------------------
Let R be a ring: a set where you can add, subtract, and multiply, with
a multiplicative identity 1. Call R DIRECTLY FINITE if, whenever two
elements a, b satisfy a b = 1, it automatically follows that b a = 1.
For finite square matrices over a field this is true: if A B = I then
B is a right inverse of A, so A is onto, so (by counting dimensions)
A is invertible and B A = I. Kaplansky's 1972 conjecture was that the
same holds in every group algebra K[G]. This paper shows the
conjecture is FALSE, even over the two-element field F_2, even when
the group G is finitely presented and has no torsion at all.
2. PREREQUISITES: GROUPS
------------------------
A group is a set G with an operation G x G -> G, written (g, h) -> g h,
and three laws:
- associativity: (g h) k = g (h k);
- identity: there is an element 1 with 1 g = g 1 = g;
- inverses: for every g there is g^-1 with g g^-1 = g^-1 g = 1.
Examples.
- The integers under addition, (Z, +).
- The nonzero reals under multiplication.
- The symmetric group S_n: all rearrangements of {1, ..., n}, under
composition.
- Z/nZ: the integers modulo n, under addition.
The order of an element g is the least positive n with g^n = 1, if
such an n exists; otherwise g has infinite order. A group is
TORSION-FREE if the only element of finite order is the identity.
A group is FINITELY PRESENTED if it can be described by finitely many
generators and finitely many equations between words in those
generators, for example
G = <x, y | x^2 = y^3 = (x y)^5 = 1>.
The phrase "x^2 = ... = 1" means: the group generated by the named
symbols, with exactly the named relations and nothing more.
3. PREREQUISITES: FIELDS, CHARACTERISTIC TWO, F_2
-------------------------------------------------
A field K is a set with + and x making it behave like the rational
numbers. A field has CHARACTERISTIC p if adding 1 to itself p times
gives 0, and p is the smallest positive integer with that property.
- The rationals Q, the reals R, the complexes C: characteristic 0.
- F_p = Z/pZ (p a prime): characteristic p, finitely many elements.
- F_2 = {0, 1} with 1 + 1 = 0: characteristic two.
In this paper the field is F_2, the smallest field there is. The fact
"1 + 1 = 0, but 1 + 1 + 1 = 1, not 0" is used constantly. Whenever we
say "because the characteristic is two", that is why. (In particular,
a plus sign is the same as a minus sign, since -1 = 1 in F_2.)
4. PREREQUISITES: GROUP RINGS (GROUP ALGEBRAS)
----------------------------------------------
Let K be a field and G a group. Write K[G] for the set of formal
finite sums
a = sum over g in G of (c_g) g,
where each c_g is in K, and only finitely many of the c_g are nonzero.
Two such sums add coefficient by coefficient. They multiply by
multiplying the group elements and collecting coefficients:
(sum c_g g) (sum d_h h) = sum over (g, h) of (c_g d_h)(g h).
The result is again a finite formal sum, because both factors had
finite support. The identity of K[G] is 1 (the group identity) with
coefficient 1. The ring K[G] is commutative exactly when G is.
Example. Let G = Z/3Z = {1, z, z^2} with z^3 = 1 and let K = F_2.
Then F_2[G] has 8 elements: every c_0 1 + c_1 z + c_2 z^2 with each
c_i in {0, 1}. Multiply:
(1 + z)(1 + z) = 1 + 2 z + z^2 = 1 + z^2 (since 2 = 0).
Fine print: K[G] is NOT the set of all functions G -> K. It is the
set of all such functions that are nonzero at only finitely many
points. Finite support is what lets the multiplication be a finite
sum, with no infinite sums anywhere.
5. WHAT "DIRECTLY FINITE" MEANS
-------------------------------
A ring R with identity is DIRECTLY FINITE if the implication
a b = 1 in R implies b a = 1 in R
holds for all a, b in R. (Kaplansky also used the term "Dedekind
finite".) For finite-dimensional linear maps over a field the
implication is automatic by dimension count. The whole point of
Kaplansky's question is that a general K[G] is infinite-dimensional:
each element is a FINITE sum, but the group G itself is usually
infinite.
A ring R is STABLY FINITE if every matrix ring Mat_n(R), n >= 1, is
directly finite. Stably finite implies directly finite. A failure of
direct finiteness is the "simplest" way an infinite-dimensional ring
can fail to behave like finite matrices.
6. KAPLANSKY'S DIRECT-FINITENESS CONJECTURE
-------------------------------------------
Conjecture (Kaplansky 1972; Fields and Rings, Part II, Section 3,
Problem 2, pp. 122-123; paraphrased). For every field K and every
group G, the group algebra K[G] is directly finite.
Before 2026 the answer was known to be YES in many cases. It is YES
in characteristic zero for EVERY group (Montgomery, 1969, gave a
short C*-algebra proof, and the same argument covers matrix rings).
It is YES when G is free-by-amenable (Ara, O'Meara, Perera 2002).
It is YES when G is sofic (Elek, Szabo 2004): sofic groups have
stably finite group algebras, over division rings (and hence fields).
The class of sofic groups is defined below in section 9.
Kaplansky's problem was one of a short list of questions he posed
about infinite group rings, and it had resisted a counterexample
for over fifty years.
7. WHAT THIS PAPER PROVES
--------------------------
The main theorem is a NO.
There is a group G that is finitely presented and torsion-free,
together with three elements a, b, c in F_2[G] satisfying
a b = 1, a c = 0, c != 0.
The three equations together force b a != 1 (see below), so
F_2[G] is not directly finite.
Moreover G admits a finite TWO-DIMENSIONAL classifying complex: a
connected CW complex with fundamental group G and contractible
universal cover.
The reason the three equations force the failure is one line of
associativity. Suppose, for contradiction, that b a = 1 as well.
Multiply a c = 0 on the left by b:
b (a c) = b 0 = 0.
(b a) c = 1 c = c.
But (b a) c = b (a c) because multiplication is associative.
So c = 0, contradicting c != 0. Thus b a cannot equal 1. The third
element c is a "certificate" that the reverse product fails; it is
never 0, but a times it is 0.
Two remarks.
(a) The field is the PRIME field F_2, not a finite extension of it.
This is stronger than the companion counterexample discussed in
section 8, which uses a larger finite field.
(b) The group is TORSION-FREE, not merely torsion. Torsion-free is
the more restrictive condition (no finite-order elements at
all, except the identity), so this version covers a strictly
harder regime.
8. HOW THIS DIFFERS FROM THE COMPANION PAPER (CHARACTERISTIC TWO,
ODD TORSION)
---------------------------------------------------------------------
A separate paper by the same author group (the "char-2 odd-torsion"
preprint, cited in the bibliography as the "characteristic two"
counterexample) also refutes Kaplansky's conjecture, but there:
- the field is a finite extension of F_2, large enough to contain
a primitive root of unity of odd prime order;
- the group contains an element of odd prime order (so it has
torsion).
This paper trades those niceties for a HARDER group (torsion-free)
and a SMALLER field (the prime field F_2 itself). It reuses the graph
and cone machinery of a third companion paper, "A Torsion-Free Group
Algebra with Zero Divisors" (cited as OpenAI2026ZD): that paper builds
a single element whose square-free product is zero; this paper
upgrades the same machinery to a one-sided inverse.
9. WHY TORSION-FREE MATTERS: THE NOT-SOFIC CONSEQUENCE
------------------------------------------------------
"Sofic" is a notion of "a group that finite computers can approximate
well". Formally: G is sofic if, for every finite set E subset G and
every eps > 0, there is a finite set Omega and a map
sigma : G -> Perm(Omega) such that
- sigma sends 1 close to the identity permutation;
- sigma is an eps-homomorphism: sigma(g h) is within eps of
sigma(g) sigma(h) for all g, h in E;
- sigma is eps-faithful: sigma(g) is far from the identity
whenever g in E is not 1.
Every finite group is sofic. Free groups are sofic. Solvable groups
are sofic. Whether every group is sofic was a big open question.
Elek and Szabo proved that sofic groups have stably finite group
algebras. Since this paper's G has a one-sided inverse that is not
two-sided, G cannot be sofic. So this construction also produces a
non-sofic group; but the point of the present paper is specifically
the direct-finiteness failure, stated over the prime field, for a
torsion-free group.
10. THE THREE INGREDIENTS: GRAPHS, CONES, PARITY
------------------------------------------------
The construction uses three pieces of machinery.
Graphs. Two finite graphs, Gamma_A and Gamma_B, whose edges are
labeled by letters from a finite alphabet T with an involution
t -> bar t (so each letter names the reversed traversal of the
same edge). Labels are chosen so that at every vertex the
outgoing labels are all distinct ("immersion").
Cones. Glue a cone on top of each connected component of the graph.
A cone is a contractible space, so every closed path inside the
graph becomes null-homotopic in the total space. The total space
X has fundamental group G, and the base rose F (one loop per
inverse label pair) generates G.
Parity. The elements a, b, c of F_2[G] are finite sums of the group
elements "read off" from vertices along paths from a fixed root.
Multiplying them produces a sum over pairs of vertices. Over
characteristic two, pairs cancel if they come in EVEN-sized
clusters. A single ODD cluster survives and produces the term
"1" in a b; all other clusters give "0" in a c. The parity is
arranged in advance by prescribing, at every vertex, the set S_x
of outgoing labels.
11. WHY THE PARITY TRICK NEEDS A SMALL "FANO PLANE" GADGET
----------------------------------------------------------
The natural way to prescribe outgoing labels is by lines of a
projective plane. Lines of the projective plane over F_128 (there
are 16513 points, 16513 lines, 129 points per line) give a very
spread-out incidence structure; vertex intersections behave
naturally in pairs, and produce the required "everything cancels"
identity for a c. But for a b we need exactly ONE uncancelled
contribution to be 1, and the plain projective plane gives none.
The fix in this paper: a set E of SEVEN extra letters, identified
with the seven points of the projective plane over F_2 (the Fano
plane). For each Fano line L (a triple of E), take the set D_L = E
minus L (four letters). The seven sets D_L, one for each Fano line,
satisfy two clean facts:
- any two distinct D_L intersect in exactly 2 letters;
- each letter lies in 4 of the 7 sets.
These two facts are exactly what is needed. They create the single
odd intersection (of the full seven-letter set with itself at the
distinguished vertex x_A) while keeping every "reverse" incidence
balanced. That single even vertex survives inside a b and gives 1;
everything else lies in even clusters and cancels.
The point is that this "Fano gadget" is not decoration. It is the
smallest combinatorial device that makes the parity calculation
produce 1 where a c produces 0.
12. WHY FINITELY PRESENTED AND TWO-DIMENSIONAL MATTERS
------------------------------------------------------
Finiteness properties keep the counterexample honest. The group G
of the theorem is not some ad-hoc infinite object cooked up by a
nonconstructive existence proof. It is:
- finitely presented: described by finitely many generators and
finitely many relations, so a computer can be handed it;
- torsion-free: no elements of order 2, 3, ... at all;
- the fundamental group of a finite two-dimensional CW complex:
so it has a finite classifying complex of dimension two.
Two-dimensional means we need only 2-cells to kill loops. This is a
strong, geometric certificate for the group; it means G has finite
cohomological dimension 2, so it inherits many good properties
(otherwise surprising the ring F_2[G] would fail direct finiteness
despite them).
13. HOW THE REST OF THIS DOCUMENT IS ORGANIZED
----------------------------------------------
PART I (this section): the question and the objects, in words.
PART II: exact statements.
PART III: the whole proof in twelve steps. Read this after PART I.
PART IV: glossary. Skim, return as needed.
PART V: the abstract.
PART VI: things to compute by hand.
PART VII: why it matters, and what stays open.
PART VIII: an index of every lemma and result.
PART IX: careful ASCII statements of the main results.
If you have an hour: read PART I, then PART III, then PART VI with
pencil and paper, then PART VII. Skip PART IX until you want the
exact form.
PART II. EXACT STATEMENTS OF THE MAIN RESULTS
================================================================================
Below are the main theorem and the major intermediate results, restated
cleanly. Throughout: F_2 = {0, 1} is the field of two elements; a
"classifying complex" for G is a connected CW complex with fundamental
group G and contractible universal cover.
Theorem 1.1 (Main theorem). There are a finitely presented torsion-free
group G and elements a, b, c in F_2[G] with
a b = 1, a c = 0, c != 0.
Consequently b a != 1. The group G admits a finite two-dimensional
classifying complex.
The role of c is the one-line proof from PART I: if b a = 1, then
c = (b a) c = b (a c) = b 0 = 0. Since c != 0, b a != 1. So the
"certificate" c turns the three equations into a failure of direct
finiteness.
Now the engine. The construction lives on labeled graphs. Fix a finite
alphabet T with a fixed-point-free involution t -> bar t. Let F be the
rose with one edge for each inverse pair {t, bar t}; oriented edges
carry the labels t and bar t. A T-LABELED GRAPH Gamma is a graph with a
label on every oriented edge, opposite orientations carrying inverse
labels, so there is a label map Gamma -> F. The map is an IMMERSION if
the labels on edges leaving any one vertex are distinct; write S_x
subset T for those outgoing labels at vertex x, and x . t for the
endpoint of the edge leaving x with label t.
A path is IMMERSED if it never immediately reverses; a closed path is
CYCLICALLY IMMERSED if this also holds at the closing turn (i.e. its
label word is freely or cyclically reduced).
Let Gamma_A, Gamma_B be finite loopless graphs immersed in the same rose,
with roots x_A in Gamma_A and x_B in Gamma_B. Put Gamma = Gamma_A
disjoint union Gamma_B. For each connected component Lambda of Gamma,
let K_Lambda be its abstract cone, and form
X = F union over (Lambda -> F) of (union of K_Lambda)
G = pi_1(X).
Each cone is attached along its label map, so a closed path in Gamma
bounds in its cone. This X is a finite CW complex of dimension at most
two. Let A' be the vertex set of the component of Gamma_A containing
x_A, and B' the vertex set of the component of Gamma_B containing x_B.
For x in A', let g_x in G be the label of any path from x_A to x, and
for y in B' let h_y be the label of any path from x_B to y.
Proposition (Parity criterion). Suppose that, for x, x' in A',
(P1) |S_x intersect S_x'| is odd for every pair except (x_A, x_A);
(P2) |S_{x_A}| is even; and
(P3) |S_x intersect S_y| is odd for every x in A' and y in B'.
Suppose also the two roots are protected:
(RP) any path in Gamma starting at x_A or x_B and ending at a
different vertex has a label that is nontrivial in G.
Then the finite scalar sums in F_2[G]
a = sum over x in A' of g_x,
b = sum over x in A' of g_x^-1,
c = sum over y in B' of h_y^-1
satisfy a b = 1, a c = 0, c != 0. Consequently b a != 1.
What each hypothesis does:
(P1)+(P2) arrange that a b has exactly one surviving odd cluster
(at (x_A, x_A)) giving the identity, and everything else
cancels in characteristic two;
(P3) makes every cluster of the A' x B' product even, so a c
cancels completely;
(RP) keeps h_y != 1 whenever y is not the root, so the
coefficient of the identity in c is exactly one, i.e.
c != 0; protection of x_B alone is what the c-part uses.
The paper keeps the symmetric condition (RP) because the
topology will prove it for both roots at once.
The proof (see PART III, step 6) uses the SIMULTANEOUS-STEP GRAPH: on
A' x A', join (x, x') to (x . t, x' . t) for every common outgoing
label t. Its degree at (x, x') is |S_x intersect S_x'|. The group
elements g_x g_x'^-1 are constant on components, and since a finite
graph has an even number of odd-degree vertices, every component with
all degrees odd contributes an even number of terms to a b (cancelling
in F_2). The one exception is the component of (x_A, x_A): it has a
single even-degree vertex, so its cardinality is odd, and its single
constant contribution is g_{x_A} g_{x_A}^-1 = 1. The same picture on
A' x B' has every degree odd, so a c cancels entirely.
Now the vertex types. Let
q = 128, v = q^2 + q + 1 = 16513, p = (q + 1) / v.
Let P and L be the points and lines of the projective plane over the
field of q elements. Concretely these are the one- and two-dimensional
subspaces of F_q^3, with incidence by containment; there are v points
and v lines, every line has q + 1 = 129 points, and every point lies on
129 lines. Two points span one line; two lines meet in one point.
Let T = P disjoint union E where E is a set of SEVEN extra letters,
identified with the seven points of the projective plane over F_2 (the
Fano plane). Let D be the family of COMPLEMENTS of the seven Fano lines
inside E; each D in D has four letters, any two distinct D's meet in
two letters, and each extra letter lies in four members of D. Pair each
extra letter with a distinct ordinary letter, and pair the remaining
ordinary letters among themselves; since v - 7 is even, this is a
fixed-point-free involution of T. The rose F has one edge per pair.
Let m run through positive integers congruent to 1 modulo 4, and put
n = v m. Split A and B into classes indexed by the 16513 projective
lines, with m vertices per class in A and m - 1 per class in B; thus
|A| = n and |B| = n - v. At every vertex x of A or B, the ORDINARY
part of S_x is its indexing line (129 points); the EXTRA part is
prescribed as follows. The distinguished vertex x_A in A gets ALL seven
extra letters. For each D in D, prescribe extra part exactly D at
a_m = ((q + 1) m - 1) / 4 vertices of A,
b_m = ((q + 1)(m - 1)) / 4 vertices of B,
keeping these demands balanced among the line classes; every remaining
vertex has empty extra part. (These counts are integers because
q + 1 = 1 = m modulo 4; the balanced split is realisable for large m.)
Proposition (Balanced types and parity). For all sufficiently large m
congruent to 1 mod 4, the sets S_x just prescribed satisfy, for every Y
in {A, B} and every t in T,
|{x in Y : t in S_x}| = |Y| p (label balance),
and their pairwise intersections satisfy exactly (P1), (P2), (P3) above.
Moreover 129 <= |S_x| <= 136 at every vertex.
Next, the edges. For each inverse pair {t, bar t}, choose a random
bijection from the t-slots to the bar-t-slots; a matched pair (x, y)
defines one edge, oriented x -> y with label t and y -> x with label
bar t. Every outgoing label occurs exactly once, so the two graphs are
immersed. Degrees lie between 129 (= d_min) and 136 (= d_*).
Write P for the resulting probability law.
Turn weights. At an interior vertex of a path with consecutive labels
t, u, the labels actually leaving the vertex are bar t and u. For a
reduced turn (u != bar t) define the weight
w(t, u) = (q + 1)^-1 if bar t and u are both ordinary,
= p if one of bar t, u is ordinary and the
other is extra,
= 1/2 if both are extra,
with w(t, bar t) = 0. For either side Y and uniformly over reduced
turns, the number of vertices of Y realising (bar t, u) equals
n p w(t, u) + O(1). For a word W = t_1 ... t_h of positive length, put
P(W) = product from i = 1 to h - 1 of w(t_i, t_{i+1}), the empty
product being 1. Then P(W) > 0 exactly when W is reduced, and
P(W^-1) = P(W).
Lemma (Decay of word weights). There is a constant delta > 0,
depending only on the alphabet and these weights, such that for all
sufficiently large integers h,
sum over W in T^h of P(W)^2 <= exp(-2 delta h).
This decay is the whole reason the random construction can be
conditioned to avoid short cycles and still keep useful bounds.
Girth conditioning. Let c_0 > 0 be small and put
L = floor(c_0 log n).
Let G_L be the event "Gamma has girth at least L" (a cycle of length
1 or 2 counts as a loop or a pair of parallel edges). The expected
number of edges on a cycle shorter than L is o(n), and a switching
argument removes such edges without creating new short ones, so G_L
is nonempty for large n; condition on it, writing P_L for the
conditional law.
Lemma (Prescriptions under girth conditioning). There is a nonnegative
sequence r_n = v p + O(d_*^(2L+2)) = o(n) with the following property.
If S is a collection of s distinct edge prescriptions with positive
P_L-probability, and e is a further prescription not in S, then
P_L(e | S) <= 1 / (n p - s - r_n) whenever n p - s - r_n > 0.
In particular, for every fixed C_1 > 0 and every specified collection
of E <= C_1 L distinct prescriptions,
P_L(all E prescriptions hold) <= (n p)^-E exp(o(L)).
Proposition (Component diameter). There is a fixed d > 0 such that,
with P_L-probability tending to 1, every connected component of
Gamma_A and of Gamma_B has diameter at most d L.
The proof uses small-set expansion: with high probability, every
vertex set S with 1 <= |S| <= rho n on either side has |N[S]| >= 2 |S|,
which forces every ball of radius R = ceil(log_2 n) + 1 to have more
than rho n vertices, and then a geodesic packing bounds every diameter
by d L.
Now the pathology exclusion. A POSITION of a path is one edge
traversal, so different positions may traverse the same edge of Gamma.
A COMPARISON is a pair of disjoint equal-length intervals of positions
whose words agree in order, or are inverse strings in reverse order.
A reduced spherical arrangement (PART IV) pairs occurrences of edges
by inverse letters in a planar way. The two theorems below say such
arrangements and such short comparison systems are both vanishingly
unlikely.
Theorem (Bounded path estimate). There is a constant epsilon > 0,
depending only on the alphabet and turn weights, with the following
property. For every integer K_0 >= 1, real C >= 1, and integer I >= 0,
the P_L-probability that Gamma contains a system with all of
(B1) at most K_0 positive-length paths, each closed and cyclically
immersed except possibly one immersed interval path starting at
x_A or x_B;
(B2) total length H between L and C L; and
(B3) at most I comparisons, satisfying the "distinct underlying
edges" rule, with b_0 <= epsilon H unpaired positions,
tends to zero as n -> infinity. Here epsilon is chosen BEFORE K_0, C,
I; the threshold in n and the rate may depend on those three.
Theorem (Absence of reduced spherical arrangements). In the random
matching model, conditioned on girth at least L, the probability that
Gamma admits a reduced spherical arrangement tends to zero as
m = 1 mod 4 tends to infinity. Consequently, for every sufficiently
large admissible m there is an outcome with girth at least L, the
component-diameter bound, and NO reduced spherical arrangement.
Finally the topological criterion, and the assembly.
Proposition (Cone criterion). Let Gamma = Gamma_A disjoint union
Gamma_B be a finite graph immersed in the rose F, with chosen roots,
and let X and G be as above. If Gamma admits no reduced spherical
arrangement, then
pi_2(X) = 0, and the root-protection condition (RP) holds.
Moreover X is a finite aspherical complex of dimension at most two, and
G is finitely presented and torsion-free.
Assembly (proof of Theorem 1.1). Use the prescribed types and let
m = 1 mod 4 tend to infinity. Condition on girth at least
L = floor(c_0 log(v m)). With probability tending to one there is an
outcome with no reduced spherical arrangement and the diameter bound;
choose one with L >= 3, so both graphs are loopless with no parallel
edges. Attach cones as above to get the finite 2-complex X and
G = pi_1(X). The cone criterion gives asphericity and root protection,
so G is finitely presented and torsion-free. The intersection parities
of the balanced-types proposition meet the hypotheses of the parity
criterion, which supplies a, b, c in F_2[G] with a b = 1, a c = 0,
c != 0. Associativity excludes b a = 1, which is Theorem 1.1.
PART III. THE SHAPE OF THE PROOF (NO DETAILS)
================================================================================
Here is the whole proof in twelve short steps. Each step is indexed
to the lemmas in PART VIII and proved in full in the original.
Step 1. Fix the alphabet and the vertex types. Set q = 128,
v = q^2 + q + 1 = 16513, p = 129 / 16513. The alphabet T is the v
points of the projective plane over F_128, plus seven extra letters E
carrying the Fano-plane structure. Every vertex gets a prescribed set
S_x of outgoing labels: size 129, 133, or 136, built from a projective
line (ordinary part) plus zero, four, or all seven extra letters. The
assignment is balanced exactly in the sense of the label-balance and
parity proposition.
Step 2. Build the random graphs. For each inverse label pair, match the
t-slots to the bar-t-slots by an independent uniform bijection. The
result is two immersed graphs Gamma_A, Gamma_B, each of minimum degree
129 and maximum degree 136. Every outgoing label at every vertex is
used exactly once.
Step 3. Condition on large girth. With L = floor(c_0 log n), the
expected number of edges on cycles shorter than L is o(n); a switching
argument deletes them without creating new ones. So "girth at least L"
is nonempty, and under the conditional law the chance of any specified
E <= C_1 L edges is at most (n p)^-E exp(o(L)).
Step 4. Show components are small. Minimum degree 129 forces
|N[S]| >= 2 |S| for every small S, so components cannot be long: with
high probability every component has diameter at most d L for a fixed
constant d.
Step 5. Attach cones. Glue a cone on every connected component of
Gamma = Gamma_A disjoint union Gamma_B. The result is a finite
two-dimensional CW complex X with fundamental group G = pi_1(X).
Every closed path inside the graph is now trivial in G, so a vertex
determines a group element by reading any path from the root.
Step 6. Run the parity calculation. The intersection parities from
Step 1 are exactly (P1), (P2), (P3) of the parity criterion. Define the
finite sums a, b, c by reading labels from the roots. Simultaneous-step
counting in characteristic two makes a b = 1 (one surviving odd
cluster at the root) and a c = 0 (all clusters even). If in addition
the roots are protected (Step 10), then every h_y with y != root
differs from 1, so c != 0, and associativity gives b a != 1.
Step 7. Reduce the geometry to one obstruction. Two facts are still
needed: X is aspherical (pi_2(X) = 0), and the two roots are protected.
The cone criterion (Step 10) says both hold exactly when Gamma admits
no REDUCED SPHERICAL ARRANGEMENT, a planar pairing of graph
occurrences by inverse letters. So the whole geometric task is to
exclude these arrangements; Steps 8 and 9 do that.
Step 8. Exclude small systems of paths (bounded path estimate). Fix
bounds K_0, C, I and the pre-chosen epsilon > 0. The probability tends
to zero that Gamma contains at most K_0 closed immersed paths (plus
possibly one rooted interval path), of total length between L and
C L, with at most I comparisons and at most epsilon H unpaired
positions. The proof is a first-moment bound, organised by the number
of times each edge is used: repeated traversals are grouped into
"multiplicity stages", and the word decay of Step 3, applied in
aligned "blocks" of length s = o(L), beats the free choices.
Step 9. Extract a small system from any arrangement (planar tools).
Any reduced spherical arrangement, however large, yields a system
covered by Step 8: an Euler-characteristic count (sum over
complementary regions) bounds the number of pairing intervals by
6 times the number of pieces; the Lipton-Tarjan planar separator
splits the pieces into clusters, and a loss-budget argument leaves one
cluster with at most epsilon H unpaired positions. Thus "no small
system" (Step 8) forces "no arrangement at all".
Step 10. Turn "no arrangement" into asphericity and protected roots
(cone criterion). If some sphere mapped into X were essential, or some
root path labelled to 1 in G, take one of MINIMAL total boundary length
and draw its cone picture. A band between two occurrences of the same
underlying edge is shortened by surgery, contradicting minimality
(this uses the homology identity that splits one essential sphere into
two capped spheres, one still essential). So neither failure can
occur, and X is aspherical with protected roots.
Step 11. Record finiteness and torsion-freeness. X is a finite
two-dimensional CW complex, so G is finitely presented. Asphericity
makes the universal cover contractible, so G has finite cohomological
dimension 2. A finite-order element would produce a prime-order cyclic
subgroup F_l, whose cohomology is nonzero in every degree (periodic
resolution), contradicting dimension 2. So G is torsion-free.
Step 12. Assemble. Choose one admissible m and one outcome satisfying
girth, diameter, and the arrangement exclusion. Its G is finitely
presented and torsion-free with a finite two-dimensional classifying
complex, and the parity criterion supplies a, b, c in F_2[G] with
a b = 1, a c = 0, c != 0. Associativity gives b a != 1, which is the
theorem. Nothing in the argument needs the field to be larger than
F_2, and nothing needs a torsion element: both features are gone.
PART IV. GLOSSARY
================================================================================
Every specialized word in the paper, defined briefly. Cross-references
are short.
Group. A set G with an associative product, an identity 1, and inverses.
Subgroup. A subset closed under product and inverse.
Order of an element. The least positive n with g^n = 1, if any; else
the element has infinite order.
Torsion-free. The only element of finite order is the identity.
Finitely presented (group). Defined by finitely many generators and
finitely many relations.
Field. A commutative ring in which every nonzero element has a
multiplicative inverse.
Characteristic p. The least positive p with 1 + 1 + ... + 1 (p times)
= 0; 0 if no such p exists.
F_2. The field {0, 1} with 1 + 1 = 0, of characteristic two. In F_2
we have -1 = 1, so plus and minus are the same, and any even number of
equal summands cancels to 0.
Group ring (group algebra) K[G]. Formal finite sums
sum c_g g with c_g in K, under convolution product. "Finite" means
only finitely many c_g are nonzero.
Directly finite (ring). a b = 1 implies b a = 1 for all a, b.
Stably finite (ring). Mat_n(R) is directly finite for every n >= 1.
Projective plane over F_q, PG(2, F_q). The set of one-dimensional
subspaces of F_q^3 ("points"), with "lines" the two-dimensional
subspaces. There are v = q^2 + q + 1 points and v lines; every line
has q + 1 points; two distinct lines meet in exactly one point. For
q = 128: v = 16513 and q + 1 = 129.
Fano plane. The projective plane PG(2, F_2): seven points, seven
lines, three points per line, three lines per point, two lines meet
in one point. Here it is used only as an index set for seven extra
letters.
Rose. A one-vertex graph with one loop for each edge of the target
alphabet (here one loop per inverse pair of labels). It is the base
space F whose fundamental group is free on the label pairs.
T-labeled graph. A graph with a label on every oriented edge, with
opposite orientations carrying inverse labels, giving a map to the
rose.
Immersion (of a labeled graph). The labels on edges leaving any one
vertex are distinct. Write S_x subset T for those outgoing labels at
vertex x, and x . t for the endpoint of the edge from x with label t.
Immersed path. A path in the graph with no immediate backtracking
(no consecutive edge followed by its reverse).
Cyclically immersed closed path. Immersed, and additionally with no
backtracking at the closing turn. Equivalently the label word is
reduced (or cyclically reduced).
Cone K_Lambda. The abstract cone on a graph Lambda: contractible space
with Lambda as its base. Gluing it onto the rose makes every closed
path in Lambda null-homotopic in the total space.
CW complex. A space built by gluing disks of increasing dimension to
the previous skeleton. "Finite 2-complex" means finitely many cells of
dimension 0, 1, 2.
Fundamental group pi_1(X). The group of loops at a basepoint, up to
homotopy; it captures the "holes" of X. Van Kampen's theorem builds
it from how spaces are glued.
Classifying complex. A connected CW complex with fundamental group G
and contractible universal cover. Such a complex exists iff G has
finite cohomological dimension equal to the complex's dimension here.
Universal cover. The simply connected covering space of X; contractible
if and only if X is aspherical.
Aspherical. All homotopy groups pi_n(X) with n >= 2 vanish, so the
universal cover is contractible. For a 2-complex, this is the same as
pi_2(X) = 0.
Girth. The length of a shortest cycle in a graph; infinite for a
forest. Here cycles of length 1 (loops) and 2 (parallel edge pairs)
also count.
Component. A connected component of a graph.
Matching (in this paper). A bijection between the t-slots Y_t and the
bar-t-slots Y_{bar t} on one side; a matched pair of slots gives one
edge. It is not the usual "graph matching".
Prescription. One matched pair in one bijection, i.e. one specified
edge. A collection of prescriptions is feasible if some tuple realises
it.
Turn. Two consecutive labels along a path. The turn is reduced if the
second label is not the inverse of the first.
Turn weight w(t, u). A positive number attached to a reduced turn,
measuring how often that turn occurs at a random vertex: it equals
(q + 1)^-1 for two ordinary labels, p for one ordinary and one extra,
1/2 for two extras, and 0 for an unreduced turn.
Word weight P(W). The product of turn weights along the consecutive
turns of a word W. It is positive exactly when W is reduced, and
symmetric under W -> W^-1.
Simultaneous-step graph. On a product A' x A' (or A' x B'), an edge
from (x, x') to (x . t, x' . t) for every common outgoing label t.
Its degree at (x, x') is |S_x intersect S_x'|; the group elements
g_x g_x'^-1 are constant along its edges.
Parity (characteristic two cancellation). Over F_2, a sum of an even
number of equal group elements is 0, and a sum of an odd number is the
element itself. So the number of terms in each constant cluster,
modulo 2, decides what survives.
Root protection. The condition (RP): any path from x_A or x_B to a
different vertex has a label that is nontrivial in G. It is exactly
what makes c != 0.
Position. One edge traversal in a path. Different positions can
traverse the same underlying edge.
Comparison. Two disjoint equal-length intervals of positions whose
words agree in order, or are inverse strings in reverse order.
Unpaired position. A position not used in any comparison; b_0 counts
them.
Multiplicity m_e. The total number of traversals of an undirected edge
e by the whole path system. It is bounded by 2(C + K_0).
Stage graph (Delta_j). The subgraph of the image graph consisting of
all edges traversed at least j times, j = 1, ..., max m_e. The stages
organise a first-moment bound in the presence of repeated traversals.
Block. A grid interval of positions of length s = L / (u R), where u
and R come from a pigeonhole selection of grid spacings; blocks are the
comparable units when two words are matched. "Full block" means a
whole interval of length s (up to O(1)).
Link. An edge of the multigraph on underlying blocks obtained by
forgetting occurrence indices of matched block occurrences.
Self-link. A link whose two occurrences belong to the same underlying
block. A translation self-link cannot respect distinct edges; a
reflection self-link is ruled out by reducedness.
Reduced spherical arrangement over Gamma. Finitely many disjoint closed
disks in an oriented sphere; every boundary reads a positive-length
cyclically immersed closed path in Gamma, except possibly one boundary
that reads an immersed interval path from a root (with the two
endpoints glued at a "break"). The edge occurrences are paired in
distinct twos by disjoint arcs outside the disks, and paired
occurrences are inverse letters on distinct underlying edges.
Digon. A complementary region of the thickened arrangement bounded by
two band-sides. A "good digon" is a disk digon whose two sides belong
to distinct bands; its two corners ("opposite gaps") must be cut
together, which is why the cut count stays controlled.
Band surgery. The topological operation that, given a band joining two
occurrences of the same underlying edge, replaces the surrounding
picture by one of shorter total boundary length. It is the reason a
minimal sphere map or minimal null-homotopy cannot have such a band.
Cone picture. A map of S^2 or D^2 into X in which the complement of
finitely many disjoint inner disks maps into the rose F, and each
inner disk maps through one abstract cone. It is the combinatorial
form of a surface map to the 2-complex.
Hurewicz, Whitehead. Two standard theorems: for a simply connected
space, the first nonzero homotopy and homology groups agree (Hurewicz);
a map between CW complexes that induces isomorphisms on all homotopy
groups is a homotopy equivalence (Whitehead). Used to pass from
"simply connected and acyclic" to "contractible".
Cohomological dimension. For a group G, the least length of a free
resolution of Z over Z G; equal to the dimension of a finite
classifying complex here, i.e. 2. A finite group F_l has infinite
cohomological dimension, which is exactly why a dimension-2 group is
torsion-free.
Lipton-Tarjan separator theorem. Any simple planar graph on s vertices
has a set of at most 2 sqrt(2 s) vertices whose removal leaves
components of at most (2/3) s vertices. Applied recursively, it gives
the "recursive planar separation" lemma.
Small-set expansion. The property that, with high probability, every
vertex set S of size at most rho n has |N[S]| >= 2 |S|, where N[S] is
S plus its neighbours. It forces balls to grow fast and hence
components to have logarithmic diameter.
Named constants used in the original.
q = 128, v = q^2 + q + 1 = 16513, p = (q + 1) / v.
d_min = 129, d_* = 136 (maximum degree).
c_0: small constant with c_0 log(2|T| / p) < 1 and 2 c_0 log d_* < 1.
L = floor(c_0 log n): the girth threshold.
d: the component-diameter constant.
delta: the word-decay rate.
epsilon: the bounded-path unpaired budget (fixed before K_0, C, I).
C_sep = 2 sqrt(2) / (1 - sqrt(2/3)): the separator constant.
D_0: the short-closure constant, at least d + 8 / (c_0 log 2) + 8.
U, eta, K_0, C, I: the extraction constants of PART III step 9, chosen
in that order.
PART V. THE ABSTRACT AND THE CONVENTIONS OF THE PAPER
================================================================================
The abstract of the original paper, quoted verbatim:
"We construct a finitely presented torsion-free counterexample to
Kaplansky's direct finiteness conjecture over the field of two
elements. The group admits a finite two-dimensional classifying
complex."
Conventions of the paper.
- Every group is written multiplicatively, with identity 1. The
two-element field is written F_2, and the same symbol denotes its
unique nonzero element when that causes no confusion.
- The map t -> bar t is a fixed-point-free involution on the label
alphabet; whenever a letter names a graph edge traversal, bar t
names the reverse traversal of the same edge.
- A graph is a one-dimensional CW complex; "length" of a path is its
number of edge traversals, counted with multiplicity.
- "Component" without qualification means a connected component of
the graph Gamma under discussion.
- All algebra is over F_2; sums are finite. Because 1 + 1 = 0, we do
not distinguish + from -, and clustering terms by group element
reduces every sum to its set of elements occurring with odd
multiplicity.
- "Above" sizes such as 129, 133, 136 come from a projective line
(129 points) plus an extra part of size 0, 4, or 7.
- Results in the original are labelled by section (Theorem 1.1,
Proposition 2, Lemma 3, ...). This companion refers to them by
their names and the files listed in PART VIII.
PART VI. WHAT A CURIOUS READER CAN ACTUALLY COMPUTE
================================================================================
Here are small calculations that make the machinery concrete. They use
nothing beyond fractions and a bit of modular arithmetic.
1. Every failure of direct finiteness comes with a certificate c.
Suppose a b = 1 but b a != 1 in some ring R. Set c = 1 - b a. Then
c != 0 and
a c = a (1 - b a) = a - a b a = a - (a b) a = a - a = 0.
So whenever direct finiteness fails, an element c with a c = 0 and
c != 0 exists for free. The theorem's c is exactly such a certificate,
constructed without first knowing b a fails. It also shows you can
never have a b = 1, a c = 0, c != 0 AND b a = 1 simultaneously.
2. Parity in characteristic two.
In F_2, g + g = 0 for any g. So a sum of k copies of the same group
element g equals g when k is odd and 0 when k is even. This is the
whole mechanism of the parity criterion: count cluster sizes mod 2.
3. The Fano plane on seven letters.
Label the seven points of PG(2, F_2) by the seven nonzero vectors of
F_2^3:
100, 010, 001, 110, 101, 011, 111.
Two points determine one line, and each line has exactly three points
(e.g. {100, 010, 110}). There are seven lines. Let E be these seven
points (the "extra letters"). For each line L, put D_L = E minus L.
Check:
- |D_L| = 7 - 3 = 4 for every line L.
- Two distinct lines meet in exactly one point, so
|D_{L1} intersect D_{L2}| = 7 - (3 + 3 - 1) = 2.
- A point belongs to exactly three lines, so it is excluded from
three D_L and included in the other four. Hence each letter lies
in 4 of the 7 sets D_L.
4. Why the Fano gadget fixes exactly the parity the theorem needs.
Inside a projective plane over F_q, the intersection of two lines is
1 point (distinct lines) or q + 1 = 129 points (the same line). Either
way, the ORDINARY part of every intersection has ODD size. Now add the
extra parts:
- empty intersect anything = 0,
- D_L intersect D_L' = 2 or 4,
- D_L intersect E = 4,
- E intersect E = 7 (ODD).
So the EXTRA part of any intersection is EVEN, except for the single
pair (x_A, x_A), where it is 7. Consequently:
- for (x, x') both in A' with (x, x') != (x_A, x_A): odd + even =
odd, which is (P1);
- |S_{x_A}| = 129 + 7 = 136, which is even, which is (P2);
- for x in A', y in B': odd + even = odd, which is (P3).
This is the entire role of the seven extra letters: they flip the
single parity at (x_A, x_A) from odd to even, and leave every other
intersection parity unchanged. Sizes come out as 129, 133 (= 129 + 4),
or 136 (= 129 + 7), all between d_min = 129 and d_* = 136.
5. Projective-plane counts.
Over F_q, PG(2, F_q) has v = q^2 + q + 1 points and the same number of
lines. Each line has q + 1 points, each point lies on q + 1 lines. For
q = 128: v = 16384 + 128 + 1 = 16513 and q + 1 = 129. The
proportion of point-letter incidences used at a given vertex is
therefore p = (q + 1) / v = 129 / 16513, about 0.0078.
6. Girth arithmetic.
A simple cycle of length k can be specified by k vertex-label choices
out of about n places, with 2 |T| |T| - 1 label choices per step.
Since the probability of a fixed edge prescription is about 1 / (n p),
the expected number of bad edges is bounded by a constant times
sum_{k=1}^{L-1} k (2 |T| / p)^k.
For q = 128, |T| = 16520 and 2 |T| / p is about 4.2 million, i.e.
roughly 2^22. If L = floor(c_0 log n) then e^{c_0 log n * 22} = n^{22 c_0};
choosing c_0 < 1/22 makes this sum o(n), so the switching argument can
clean out every short cycle. The paper's constants (2 c_0 log d_* < 1,
etc.) are slightly more careful versions of this count.
7. Why small sets expand (sketch).
If a set S has k vertices, every x in S has degree at least 129, so
at least 129 k / 2 distinct edges touch S. All their endpoints lie in
N[S]. A union bound over choices of S and of those edges, using the
(n p)^-E prescription bound, shows that with high probability no S of
size at most rho n has |N[S]| < 2 |S|. (In the original this count is
eq:expansion-probability: choose 2 k vertices for N[S] and about 65 k
edges, giving (C_3 (k/n)^62.5)^k summed over k, which is o(1).) The
inequality |N[S]| >= 2 |S| doubles every closed ball, so every
component has diameter at most d L with d fixed.
PART VII. WHY THIS MATTERS
================================================================================
1. It settles Kaplansky's direct-finiteness conjecture in characteristic
two. Kaplansky (Fields and Rings, 1972, Part II, Section 3, Problem 2)
asked whether a b = 1 implies b a = 1 in every group algebra. This
paper answers NO over the field of two elements, with the group
finitely presented, torsion-free, and admitting a finite two-dimensional
classifying complex. That is close to the strongest possible negative
answer: the field is the smallest field there is, and the group has no
torsion at all.
2. What was already known before 2026.
- Characteristic zero: direct finiteness holds for every group
(Montgomery 1969, and the same operator-algebra argument gives
stable finiteness, i.e. it holds in every matrix size).
- Free-by-amenable groups: stably finite over every division ring
(Ara, O'Meara, Perera 2002).
- Sofic groups: stably finite (Elek, Szabo 2004). Since every
finitely presented group with a "nice" recursive presentation is
sofic (residually finite, virtually free, linear, ...), soficity
had covered essentially every finitely presented example where
direct finiteness was tested.
- Companion preprints: the zero-divisor paper (OpenAI2026ZD) built
ab = 0, a, b != 0 in F_2[G] with G finitely presented, torsion-
free; the char-2 odd-torsion paper (OpenAI2026DF) built a
one-sided inverse, but the field was a finite extension of F_2 and
the group had odd-order torsion.
This paper combines the "prime field" strength of the first with the
"one-sided inverse" strength of the second: F_2 and torsion-free
simultaneously.
3. Consequence for soficity. Elek and Szabo proved that every sofic
group has stably finite group algebra. A directly-finite failure rules
that out, so the group G of Theorem 1.1 is NOT sofic. This is a new
finitely presented non-sofic group, though the paper's focus is direct
finiteness, not soficity.
4. Why torsion-free matters. Groups of finite cohomological dimension
(here 2) are often the best-behaved groups in the theory. This paper
shows that even at cohomological dimension 2, over the smallest field,
direct finiteness can fail. Any suggestion that "torsion is the
obstruction to direct finiteness" or that "finite-dimensional group
cohomology is enough" is thereby refuted.
5. Why the prime field F_2 matters. Earlier counterexamples (in the
companion paper) needed a larger finite field to contain a primitive
l-th root of unity. The field here is {0, 1}, with nothing extra
available. This removes the last algebraic convenience people might
have hoped was essential.
6. Concrete open questions that this result raises.
- Is a direct-finiteness failure possible in characteristic zero?
(Known NO for all groups; the operator-algebra argument is
unconditional.)
- Which finitely presented groups have F_2[G] stably finite?
Stably finite is a stronger property than directly finite; the
paper produces a directly-finite failure, but not necessarily a
stable-finiteness failure.
- Are there lower-complexity (e.g., cohomological dimension 2 and
residually finite) direct-finiteness failures? Here the group is
not residually finite, since a residually finite group that is
finitely presented is sofic.
- What is the smallest number of generators and relations for a
directly-finiteness failure over F_2? This paper does not
optimise that.
- Does the same parity mechanism produce a zero-divisor a b = 0,
a, b != 0, over a smaller field or in lower dimension? Partly
answered by the companion zero-divisor paper.
7. What a high school reader should take away.
- "Same object, two-sided inverse" is not automatic in infinite
settings.
- A failure of direct finiteness can be certified by one extra
element c satisfying a c = 0, c != 0; if that c were zero, no
failure would be possible.
- Characteristic two (i.e. "1 + 1 = 0") turns a counting
problem into algebra: count the parity of clusters, and a parity
prescription builds the desired product.
- The construction is combinatorial: finite graphs, random
matchings, and a small amount of linear-algebra-free bookkeeping.
The topology involved is 2-dimensional, not higher.
- Big theorems are not needed: "the graph has large girth" and
"the plane separator lemma" suffice. The non-obvious ingredient
is the little seven-letter gadget (Fano plane), which fixes the
one needed parity without disturbing the rest.
PART VIII. THE LEMMA AND RESULT INDEX OF THE ORIGINAL PAPER
================================================================================
The original paper is organised as A-Torsion-Free-Group-Algebra-That-
Is-Not-Directly-Finite// plus section files. Named results, in
alphabetical order of their labels, with the file that proves them and
a one-line description.
theorem thm:main introduction.tex
Main theorem: finitely presented, torsion-free G and
a, b, c in F_2[G] with a b = 1, a c = 0, c != 0; hence
b a != 1; and G has a finite 2-dimensional classifying complex.
proposition prop:parity algebra.tex
Parity criterion: graphs immersed in one rose, intersection
parities (P1), (P2), (P3) plus root protection, yield the
finite sums a, b, c with a b = 1, a c = 0, c != 0.
proposition prop:types random.tex
Balanced types and parity: label balance and (P1)-(P3) for
the projective-plane + Fano extra-letter types.
lemma lem:word-decay random.tex
decay of P(W)^2: sum_{W in T^h} P(W)^2 <= exp(-2 delta h).
lemma lem:prescriptions random.tex
Prescriptions under girth conditioning: P_L(e | S) <=
1 / (n p - s - r_n); for E <= C_1 L prescriptions, at most
(n p)^-E exp(o(L)).
proposition prop:diameter random.tex
Component diameter: with P_L-probability -> 1, every
component of Gamma_A, Gamma_B has diameter at most d L.
theorem thm:bounded-patterns patterns.tex
Bounded path estimate: probability -> 0 of any system of at
most K_0 cyclically immersed paths (plus possibly one rooted
interval path), total length L <= H <= C L, at most I
comparisons, fewer than epsilon H unpaired positions.
lemma lem:bounded-image-rank patterns.tex
Bounded cycle rank of a girth-L graph with at most C L edges.
lemma lem:multiplicity-incidence patterns.tex
Stage-incidence identities organising repeated edge
traversals.
lemma lem:link-word-counts patterns.tex
Word counts per link component, with self-links and the
multiplicity inequality 2 z_U <= S_U + b_U when no self-link.
definition def:reduced-arrangement planar.tex
Reduced spherical arrangement over Gamma: disks in a sphere,
inverse-letter occurrence pairings on distinct underlying
edges, at most one break.
theorem thm:no-arrangements planar.tex
Absence of reduced spherical arrangements with probability
-> 1 under girth conditioning.
lemma lem:closure planar.tex
Short closures: every immersed segment is extendable to a
cyclically immersed closed path, appending at most D_0 L
edges, with D_0 >= d + 8 / (c_0 log 2) + 8.
lemma lem:paired-intervals planar.tex
Paired intervals: an arrangement on N' pieces has at most
6 N' disjoint interval pairs.
lemma lem:separator planar.tex
Recursive planar separation: a set of at most
C_sep N'/sqrt(K_0) vertices leaves every component with at
most K_0 vertices, where C_sep = 2 sqrt 2 / (1 - sqrt(2/3)).
proposition prop:cone-criterion topology.tex
Cone criterion: if Gamma has no reduced spherical arrangement,
then pi_2(X) = 0 and root protection (RP) holds; X is finite
aspherical 2-dimensional; G is finitely presented and
torsion-free.
lemma lem:cone-pictures topology.tex
Every sphere or disk map to X is homotopic to a cone picture;
with prescribed boundary when it is a disk.
proof assembly.tex
Completion of Theorem 1.1: choose one finite pair of graphs
satisfying types, diameter, no arrangements; assemble.
Figures in the original (the "why" of the geometric argument).
A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite//blocks.tex
figure fig:aligned-blocks: nearly aligned blocks and the
occurrence/underlying-block distinction.
...//digon.tex
figure fig:good-digon: a good gap and its opposite gap between
two distinct bands; the disk digon Q.
...//exterior-surgery.tex
figure fig:exterior-surgery: a band returning to the exterior
disk, replaced by PR with the same graph endpoints.
...//inner-surgery.tex
figure fig:inner-surgery: an inner cone disk plus a returning band
splitting into two capped spheres, with the homology identity
[f] = [f_1] + [f_2].
Named equations in the original (by LABEL).
algebra.tex eq:cone-complex, eq:path-step, eq:parity-assumptions,
eq:root-protection, eq:scalar-elements.
random.tex eq:extra-counts, eq:label-balance, eq:type-parity,
eq:degree-bound, eq:turn-weights, eq:turn-count,
eq:word-weight, eq:word-decay, eq:girth-scale,
eq:bad-edges, eq:one-prescription, eq:short-prescriptions,
eq:small-set-expansion, eq:expansion-probability.
patterns.tex eq:distinct-comparison-edges, eq:bounded-multiplicity,
eq:stage-incidence, eq:comparison-affine-map,
eq:block-scale, eq:block-unmatched-budget,
eq:link-component-parameters, eq:component-multiplicity,
eq:word-bin-bounds, eq:pattern-entropy,
eq:fixed-stage-expectation, eq:stage-label-count,
eq:stage-markov, eq:stage-product, eq:epsilon-choice.
planar.tex eq:arrangement-euler, eq:non-good-gaps, eq:added-length,
eq:extraction-piece-count, eq:deleted-occurrences,
eq:extraction-constants, eq:extraction-losses,
eq:cluster-budgets.
topology.tex eq:inner-surgery-words, eq:exterior-surgery-words,
eq:sphere-surgery-homology.
PART IX. EXACT STATEMENTS OF EVERY RESULT (VERBATIM, FROM THE ORIGINAL)
================================================================================
Lightly retypeset for ASCII: LaTeX macros were expanded, and slanted
notation was flattened. The mathematical content is unchanged.
Theorem 1.1 (Main theorem). There exist a finitely presented torsion-
free group G and elements a, b, c in F_2[G] such that
a b = 1, a c = 0, c != 0.
Consequently b a != 1. The group G admits a finite two-dimensional
classifying complex.
Proposition (Parity criterion). Let Gamma_A, Gamma_B be finite loopless
graphs immersed in the same rose, with roots x_A in A' and x_B in B',
and cone group G as above. Suppose that
|S_x intersect S_x'| = 1 (mod 2) for x, x' in A',
(x, x') != (x_A, x_A),
|S_{x_A}| = 0 (mod 2),
|S_x intersect S_y| = 1 (mod 2) for x in A', y in B'.
Suppose also that the roots are protected in the following sense:
if a path in Gamma starts at x_A or x_B and ends at a different
vertex, its label is nontrivial in G.
Then the finite scalar sums
a = sum over x in A' of g_x,
b = sum over x in A' of g_x^-1,
c = sum over y in B' of h_y^-1,
in F_2[G], satisfy a b = 1, a c = 0, and c != 0. Consequently b a != 1.
Proposition (Balanced types and parity). For all sufficiently large m
with m = 1 (mod 4), the prescribed sets S_x subset T satisfy:
|{x in Y : t in S_x}| = |Y| p for every Y in {A, B}, t in T;
|S_x intersect S_x'| = 1 (mod 2) for x, x' in A,
(x, x') != (x_A, x_A);
|S_{x_A}| = 0 (mod 2);
|S_x intersect S_y| = 1 (mod 2) for x in A, y in B.
Moreover 129 <= |S_x| <= 136 at every vertex.
Lemma (Decay of word weights). There is a constant delta > 0,
depending only on the prescribed alphabet and weights, such that for
all sufficiently large integers h,
sum over W in T^h of P(W)^2 <= exp(-2 delta h).
Lemma (Prescriptions under girth conditioning). There is a nonnegative
sequence r_n = v p + O(d_*^(2L+2)) = o(n) with the following property.
If S is a collection of s distinct prescriptions with positive
P_L-probability, and e is a further prescription not in S, then
P_L(e | S) <= 1 / (n p - s - r_n) whenever n p - s - r_n > 0.
In particular, for every fixed C_1 > 0, a specified collection of
E <= C_1 L distinct prescriptions satisfies
P_L(all E prescriptions hold) <= (n p)^-E exp(o(L)).
Proposition (Component diameter). There is a fixed constant d > 0
such that, with P_L-probability tending to one as m -> infinity, every
connected component of Gamma_A and of Gamma_B has diameter at most
d L.
Theorem (Bounded path estimate). There is a constant epsilon > 0,
depending only on the fixed alphabet and turn weights, with the
following property. For every integer K_0 >= 1, real number C >= 1,
and integer I >= 0, the probability under P_L that Gamma contains a
system satisfying all of the following conditions tends to zero as
n -> infinity:
(1) there are at most K_0 positive-length paths, each closed and
cyclically immersed, except possibly one immersed interval
path whose initial vertex is x_A or x_B;
(2) the total length satisfies L <= H <= C L;
(3) there are at most I comparisons, satisfying the
"distinct underlying undirected edges" rule, and
b_0 <= epsilon H unpaired positions.
In particular epsilon is chosen before K_0, C, I; the threshold in n
and the rate of convergence may depend on those three bounds.
Definition (Reduced spherical arrangement over Gamma). A reduced
spherical arrangement over Gamma consists of finitely many pairwise
disjoint piecewise linear closed disks in an oriented sphere, together
with the following data. Each boundary reads a positive-length
cyclically immersed closed path in Gamma, except possibly one boundary
that reads a positive-length immersed interval path starting at x_A or
x_B, with its two endpoints represented at a single marked boundary
point, the break. No immersion condition is imposed across the break.
At least one ordinary, cyclically immersed boundary is required. The
paths are read in the boundary orientations induced by the disks.
Their edge occurrences are paired in distinct twos by mutually disjoint
piecewise linear arcs. The interiors of the arcs lie outside the
disks, and each occurrence contains exactly one arc endpoint in its
interior. Paired occurrences have inverse signed letters and use
distinct underlying undirected edges of Gamma.
Theorem (Absence of reduced spherical arrangements). In the random
matching model, conditioned on girth at least L, the probability that
Gamma admits a reduced spherical arrangement tends to zero as
m = 1 (mod 4) tends to infinity. Consequently, for every sufficiently
large admissible m there is a matching outcome with girth at least L,
the component-diameter bound, and no reduced spherical arrangement.
Lemma (Short closures). There is a fixed integer D_0, depending only
on d and c_0, with the following property for all sufficiently large
admissible n. If every component of Gamma has diameter at most d L,
then every positive-length immersed path segment in Gamma extends to a
cyclically immersed closed path by appending at most D_0 L edges. The
original segment is retained without cancellation. One may take any
integer D_0 >= d + 8 / (c_0 log 2) + 8.
Lemma (Paired intervals). Suppose a reduced spherical arrangement with
N_0 disk boundaries is divided into N' >= N_0 pieces by cutting its
paths into consecutive segments. An intact path counts as one piece,
and no piece crosses the exceptional break. Then all original
occurrence pairings can be represented by at most 6 N' disjoint pairs
of intervals. Each interval is contained in one designated piece, and
each pair compares inverse strings in reverse boundary order. The
intervals can also be required not to cross one prescribed indexing gap
between occurrences on each original path.
Lemma (Recursive planar separation). Let
C_sep = 2 sqrt(2) / (1 - sqrt(2/3)).
For every integer K_0 >= 1, a finite planar multigraph with N' vertices
has a set of at most C_sep N' / sqrt(K_0) vertices whose deletion
leaves every connected component with at most K_0 vertices.
Proposition (Cone criterion). Let Gamma = Gamma_A disjoint union
Gamma_B be a finite graph immersed in the rose F, with chosen roots,
and let X and G be defined as above. If Gamma admits no reduced
spherical arrangement as in the definition, then
pi_2(X) = 0, and the root-protection condition holds.
Moreover X is a finite aspherical complex of dimension at most two,
and G is finitely presented and torsion-free.
Lemma (Cone pictures). Every map S^2 -> X is homotopic to a cone
picture. Every disk map whose boundary is the label of a specified
edge path in Gamma is homotopic relative to that boundary to a cone
picture.
END OF COMPANION VERSION
================================================================================
How to use this document.
- To understand WHAT the theorem says, read PART I and PART II, then
PART IX for exact statements.
- To understand WHY it is true, read PART I and PART III. Pencil and
paper: PART VI.
- To learn WHY it matters, read PART VII.
- To check terminology, PART IV.
- To look up a specific result, PART VIII.
The original paper (torsion_free_group_not_directly_finite_original.txt,
compiled from A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-
Finite//*.tex) contains the full proofs and every estimate. This
companion is not a substitute for the original: it is the map, not the
territory. Where this document says a lemma "is proved by a first-
moment bound", "is proved by a switching argument", or "is proved by
band surgery", the original carries every constant and every inequality.
Nothing in this companion is meant to replace careful reading of the
original. It is meant to make that careful reading possible for a
reader who does not yet know the vocabulary.