==== TABLE OF CONTENTS ====
An-infinite-finitely-presented-residually-finite-2-group//introduction.tex
An-infinite-finitely-presented-residually-finite-2-group//ring-phases.tex
An-infinite-finitely-presented-residually-finite-2-group//steinberg.tex
An-infinite-finitely-presented-residually-finite-2-group//shifts.tex
An-infinite-finitely-presented-residually-finite-2-group//long-entries.tex
An-infinite-finitely-presented-residually-finite-2-group//detection.tex
An-infinite-finitely-presented-residually-finite-2-group//stability.tex
An-infinite-finitely-presented-residually-finite-2-group//construction.tex
An-infinite-finitely-presented-residually-finite-2-group//references.tex
===========================

==== BEGIN An-infinite-finitely-presented-residually-finite-2-group//introduction.tex ====
\section{Introduction}\label{sec:introduction}

We prove the following statement.

\begin{theorem}[Main theorem]\label{thm:main}
There exists an infinite, residually finite group with an ordinary
finite presentation in which every element has finite 2-power order.
\end{theorem}

The rest of this section fixes terminology, surveys the relevant
history, and outlines the argument. Readers new to group theory may
skip straight to the short dictionary at the end of the section.

\subsection*{The terms of the statement}

A \emph{group} is a set $G$ with an associative binary operation, an
identity element $1$, and inverses. It is \emph{finitely generated}
if there is a finite set whose products (and inverses) give all of
$G$. An \emph{ordinary finite presentation} consists of finitely many
generators $s_1,\dots,s_m$ and finitely many relations $w_k=1$; the
group is the quotient of the free group on the generators by the
normal closure of the words $w_k$. A group is \emph{periodic} if
each of its elements has finite order. It is a \emph{2-group} if
each element has order $2^a$ for some integer $a\ge0$; the group
itself is allowed to be infinite. It is \emph{residually finite} if,
for every non-identity element $g$, there is a homomorphism from the
group to a finite group whose value at $g$ is non-identity. Finite
presentation is taken in the category of abstract discrete groups,
not in a variety of groups or a pro-2 category.

\subsection*{Why the statement is surprising}

Burnside \cite{Burnside1902} asked in 1902 whether finitely generated
groups of bounded exponent must be finite. Novikov and Adian
\cite{NovikovAdian1968} answered no, constructing infinite
finitely generated groups of small bounded exponent. Those groups
are notoriously hard to describe, and they cannot be residually
finite: Zel'manov's solution of the restricted Burnside problem for
2-groups \cite[Theorem~2]{Zelmanov1992} implies that a finitely
generated residually finite group whose element orders are bounded
by a fixed $2^a$ is finite. Theorem~\ref{thm:main} sits at the other
extreme: the group is infinite and periodic, and its finite quotients
are still plentiful enough to separate all of its elements. Its
element orders are necessarily unbounded; and, because every element
has 2-power order, Cauchy's theorem shows that every finite quotient
is itself a finite 2-group.

\subsection*{History}

Golod \cite[Example~2]{Golod1964}, using the dimension criterion of
Golod and Shafarevich \cite{GolodShafarevich1964}, produced the
first infinite finitely generated residually finite $p$-groups. His
mechanism anticipates ours in outline: nilpotence produces finite
$p$-power orders, while a grading separates elements in finite
quotients. Grigorchuk \cite{Grigorchuk1980} gave the first concrete
example, an infinite finitely generated residually finite 2-group
which is not finitely presented. Neither construction produces an
ordinary finite presentation of the whole group.

Whether a finitely presented infinite periodic group exists is
recorded as an open question in
\cite[p.~45]{OlshanskiiSapir2003} and again in
\cite[Section~2.2(IX)]{HaettelOsajda2025}. The companion paper
\cite{Companion} answers it affirmatively, by constructing an
infinite finitely presented periodic group as a Steinberg group over
a finitely presented graded algebra with a matrix-nil positive-degree
ideal. The companion's group is not known to be residually finite.
The present paper strengthens the companion in two ways: it proves
residual finiteness, and it produces a finite-index subgroup whose
every element has 2-power order. The companion's algebra is an
essential input, quoted precisely at the start of
Section~\ref{sec:ring-phases}.

\subsection*{The group and its finite quotients}

The group in Theorem~\ref{thm:main} arises as follows. The companion
supplies a unital associative $\F_2$-algebra
\[
 R=\F_2\oplus I,\qquad I=\bigoplus_{d\ge1}R_d,
\]
finitely presented, graded by nonnegative integers, with every $R_d$
finite-dimensional, and with the strong nilpotence property that
every finite square matrix with entries in $I$ is nilpotent. We
consider the Steinberg group
\[
 \Gamma=\St_{12}(R),
\]
generated by symbols $x_{ij}(a)$, $a\in R$, $i\ne j$, subject to
the algebraic relations stated in Section~\ref{sec:steinberg}. Its
matrix image is the elementary matrix group $\E_{12}(R)$ generated
by the elementary matrices $e_{ij}(a)$, and $\Gamma$ has an
ordinary finite presentation by \cite[Theorem~4.1]{Companion} (see
also \cite[Theorem~3]{KrsticMcCool}).

The finite quotients of $\Gamma$ come from degree truncations of
$R$. For $d\ge1$ put
\[
 R_{\ge d}=\bigoplus_{e\ge d}R_e,\qquad B_d=R/R_{\ge d}.
\]
Each $B_d$ is finite: it is a direct sum of finitely many
finite-dimensional $\F_2$-vector spaces. Moreover
$B_d=\F_2\oplus I_d$ with $I_d^d=0$, so the companion's nilpotence
hypothesis still applies, and we obtain finite quotients
\[
 \Gamma\longrightarrow\St_{12}(B_d).
\]
The heart of the paper is the following detection statement: for
every non-identity $g\in\Gamma$, some $d\ge1$ makes the image of
$g$ in $\St_{12}(B_d)$ non-identity.

\subsection*{How detection works}

Detection is proved in two nested layers. The outer layer,
Section~\ref{sec:stability}, is a stability theorem: adjoining
coordinates does not enlarge the kernel. Concretely, the standard
map $\St_{12}(R)\to\St(R)$ to the stable Steinberg group is
injective. This lets detection be carried out in the stable group
$\St(R)$, where the proof may freely use additional coordinates.

The inner layer, Sections~\ref{sec:shifts}--\ref{sec:detection},
separates points of the stable group. Put
\[
 \E(R)=\varinjlim_n\E_n(R),\qquad
 J(R)=\ker\bigl(\St(R)\longrightarrow\E(R)\bigr).
\]
The kernel $J(R)$ is central in $\St(R)$
(Lemma~\ref{lem:stable-center}). Where $\E(R)$ is concerned,
separation is immediate: $g\notin J(R)$ already has a non-identity
matrix image over $R$, which some finite truncation sees. The real
content is detection inside the central kernel $J(R)$.

That is done by two cyclic-shift computations. Let $C_b$ be the
$b\times b$ cyclic permutation matrix over $\F_2$. Applying the
graded substitution $t\mapsto C_b$ to a root word for $z\in J(R)$
produces an element $W_z(C_b)\in J(R)$. The two computations are:

\begin{itemize}\itemsep0.2em
\item \emph{Power of two.} If $b$ is a power of two and large
  enough, then $C_b$ is conjugate in $\GL_b(\F_2)$ to an upper
  triangular matrix with all diagonal entries $1$. Triangular
  evaluation and coordinate permutations then give $W_z(C_b)=z^b$
  whenever $b$ is large enough that the wrap of $C_b$ can be
  removed (Lemma~\ref{lem:wrap}); on the other hand
  Lemma~\ref{lem:cyclic-vanishing} gives $W_z(C_b)=1$ whenever $b$
  is large enough. Hence $z^{2^a}=1$ for some $a\ge0$.
\item \emph{Odd power.} If $L$ is odd and $z$ is missed by every
  finite truncation, then a representative of $z$ with very long
  entries exists, and the \emph{phases} of long words in $R$ let us
  transport root endpoints to coordinates labeled by their phases.
  This gives $W_z(C_L)=z^L$ (Proposition~\ref{prop:odd-power}).
  Combined with the vanishing $W_z(C_L)=1$ for large enough odd $L$
  (Lemma~\ref{lem:cyclic-vanishing}), we obtain $z^L=1$.
\end{itemize}

Since $2^a$ and $L$ are coprime, $z=1$. That completes detection of
$J(R)$, hence of $\St(R)$, hence (by stability) of $\Gamma$.

\subsection*{The 2-group}

Once $\Gamma$ is residually finite, we pass to
\[
 G=\ker\bigl(\St_{12}(R)\longrightarrow\St_{12}(\F_2)\bigr).
\]
This is finite index in $\Gamma$ because $\St_{12}(\F_2)$ is finite
(the case $d=1$ of the finite-quotient construction above). It is
infinite and finitely presented by the Reidemeister--Schreier
theorem \cite[Chapter~II, Section~4]{LyndonSchupp}, and it inherits
residual finiteness from $\Gamma$. Every element has finite 2-power
order. Indeed, if $g\in G$ has matrix image $\Id+A$ with $A\in
M_{12}(I)$ and $A^r=0$, choose $a$ with $2^a\ge r$. In
characteristic two,
\[
 (\Id+A)^{2^a}=\Id+A^{2^a}=\Id,
\]
so $g^{2^a}\in J_{12}(R)$. Its stable image lies in $J(R)$ and
reduces to $1$ in $\St(\F_2)$, so
Proposition~\ref{prop:relative-torsion} gives an integer $b\ge0$
with $g^{2^b}=1$ stably. Injectivity of
$\St_{12}(R)\to\St(R)$ then gives $g^{2^{a+b}}=1$ in $\Gamma$, hence
in $G$.

\subsection*{A dictionary for readers new to the subject}

Readers comfortable with the terms above may skip this list.

\begin{itemize}\itemsep0.2em
\item \textbf{Group.} A set with an associative multiplication, an
  identity, and inverses.
\item \textbf{Generator.} An element whose products (and their
  inverses) produce the whole group.
\item \textbf{Relation.} An equation between two products of
  generators, usually written $w=1$ for a word $w$.
\item \textbf{Finite presentation.} Finitely many generators and
  finitely many relations.
\item \textbf{Order.} The smallest $n\ge1$ with $g^n=1$, if it
  exists.
\item \textbf{Periodic.} Every element has finite order.
\item \textbf{$p$-group.} Every element has order a power of the
  prime $p$; here $p=2$.
\item \textbf{Residually finite.} For every non-identity $g$, some
  homomorphism to a finite group sends $g$ to something
  non-identity.
\item \textbf{Matrix and Steinberg groups.} $\E_n(R)$ is generated
  by the elementary matrices $e_{ij}(a)$ (identity plus $a$ in one
  off-diagonal position). $\St_n(R)$ is generated by formal symbols
  $x_{ij}(a)$, subject to the same algebraic relations; it covers
  $\E_n(R)$ with kernel $J_n(R)$.
\end{itemize}

For a reader encountering these objects for the first time, it may
help to picture $\E_n(R)$ as the group generated by very small
matrix tweaks, and $\St_n(R)$ as its well-behaved formal lift:
symbols $x_{ij}(a)$ know only the algebraic structure of these
tweaks, not the particular matrices they happen to be.

\subsection*{How to read the rest}

Section~\ref{sec:ring-phases} records the companion algebra's word
presentation and extracts the phases of long words.
Section~\ref{sec:steinberg} introduces the Steinberg group and the
faithful-strip lemmas that the rest of the paper uses without
comment. Section~\ref{sec:shifts} proves the power-of-two
computation and the vanishing of cyclic shifts.
Section~\ref{sec:long-entries} controls opposite roots for long
entries. Section~\ref{sec:detection} combines both computations
into detection by finite degree truncations.
Section~\ref{sec:stability} transfers detection to fixed rank $12$.
Section~\ref{sec:construction} constructs the finite quotients
explicitly and proves Theorem~\ref{thm:main}.

All arguments establishing the phase, stabilization, and detection
claims are included below. From the companion we quote only the
algebra construction and the finite-rank elementary facts needed to
apply it, with hypotheses stated at the points of use.
==== END An-infinite-finitely-presented-residually-finite-2-group//introduction.tex ====