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\hypersetup{
  pdftitle={Uniform Kazhdan Monsters and Schmidt's Property M},
  pdfauthor={Mehdi Moradi},
  pdfsubject={Schmidt's Property M, uniform Kazhdan constants, and quasifinite groups},
  pdfkeywords={MSC2020 37A25, 22D55, 20F65; mild mixing; property M; uniform Kazhdan constants}
}

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\title{Uniform Kazhdan Monsters and Schmidt's Property M}
\author{Mehdi Moradi\\[0.3em]
  \small Department of Mathematics, University of Toronto\\
  \small Toronto, Ontario, Canada\\
  \small\href{mailto:mehdi.moradi@utoronto.ca}%
    {\texttt{mehdi.moradi@utoronto.ca}}\\
  \small\href{https://orcid.org/0000-0001-9329-5634}{ORCID: 0000-0001-9329-5634}}
%\date{Hexagon review copy -- 7 October 2026}

\begin{document}

\maketitle

\begin{abstract}
Let \(G\) be an infinite finitely generated group whose proper subgroups are
finite.  We prove that a positive Kazhdan constant uniform over all finite
generating sets forces every unit vector in every representation without
invariant vectors to have a cofinite displacement gap.  For ergodic
probability-measure-preserving actions this gives
\[
 \liminf_{g\to\infty}\mu(gA\mathbin\triangle A)
 \geq \ku(G)^2\mu(A)(1-\mu(A)),
\]
and hence Schmidt's Property~\(M\).  A tensor-square argument also gives a
uniform gap from modulus one for diagonal matrix coefficients.  Conditional
on the graded-diagram package underlying the Osin--Sonkin construction, the
theorem shows that their infinite finitely generated simple ICC group of
bounded exponent has Property~\(M\).  We include a corrected, constant-tracked
proof of the Osin--Sonkin deduction from that package and conclude with
questions about Ozawa's quasifinite Kazhdan quotients.
\end{abstract}

{\small\noindent
\textit{2020 Mathematics Subject Classification.}
Primary 37A25; Secondary 22D55, 20F65.\par}

\section{Introduction and context}\label{sec:introduction}

Furstenberg and Weiss introduced mild mixing through the multiplier problem,
and Schmidt and Walters developed it for locally compact group actions
\cite{FurstenbergWeiss1978,SchmidtWalters1982}.  Related foundations and
later formulations appear in
\cite{Dye1965,AaronsonLinWeiss1979,Walters1983,
BergelsonRosenblatt1988,HawkinsSilva1998,Glasner2003}.  Schmidt associated to
a noncompact locally compact second countable group the properties \(W\),
\(M\), and \(S\): every ergodic probability-measure-preserving action is,
respectively, weakly, mildly, or strongly mixing, with
\(S\Rightarrow M\Rightarrow W\) \cite{Schmidt1984}.  He also proved that an
infinite discrete \(M\)-group is finitely generated and that every proper
subgroup is finite; it is perfect and simple modulo a finite center
\cite[Lemma~6.2 and Theorem~6.3]{Schmidt1984}.\footnote{The conclusion of
Theorem~6.3 is printed with \emph{every subgroup} where the assumptions and
proof require \emph{every proper subgroup}.}  Schmidt knew no discrete
example and pointed to newly constructed simple groups without infinite
proper subgroups as candidates; the absence of a known countably infinite
example was still noted in \cite[pp.~83--84]{Kechris2010}.

Property~\(M\) is distinct from two neighboring rigidity notions.  The
Connes--Weiss characterization of Property~\((T)\) concerns strong
ergodicity: it rules out nontrivial sequences of sets that are asymptotically
invariant under every fixed group element
\cite{ConnesWeiss1980,Schmidt1981,JonesSchmidt1987}.  Failure of mild mixing
instead fixes one nontrivial set and lets the group element escape to
infinity.  At the stronger end, Property~\(S\) is the Howe--Moore property:
all coefficients of representations without invariant vectors vanish at
infinity \cite{HoweMoore1979,Schmidt1984,CluckersEtAl2011}.  Mayer's
representation-theoretic condition also called mild mixing is stronger than
the Schmidt--Walters notion and concerns a different classification problem
\cite{Mayer1997}.
Cluckers--Cornulier--Louvet--Tessera--Valette showed, more specifically, that
an infinite discrete Howe--Moore group is finitely generated and
finite-by-(quasifinite simple), while emphasizing how little was known about
representations of quasifinite groups
\cite[Lemma~3.4 and Remark~3.5]{CluckersEtAl2011}.

The relevant groups originate in small-cancellation constructions.
Ol'shanskii produced finitely generated groups with finite cyclic proper
subgroups, and later work produced periodic and quasifinite quotients of
hyperbolic groups
\cite{Olshanskii1979,Gromov1987,Olshanskii1992,Olshanskii1993,
IvanovOlshanskii1996}.  Ozawa used this machinery to obtain uncountably many
quasifinite ICC quotients, which are Kazhdan when the hyperbolic source is
Kazhdan \cite{Ozawa2004}.  Osin and Sonkin introduced Property~\(U\) and
used Semenov's graded small-cancellation construction to produce the first
infinite groups whose Kazhdan constants are bounded away from zero over all
finite generating sets \cite{Semenov1993,OsinSonkin2006}.  Further periodic groups with
Property~\(U\) were constructed in \cite{Atabekyan2008,Rostami2008}, but
those papers do not assert Property~\((T)\).

For context, Lubotzky asked whether every Kazhdan group has such a uniform
lower bound \cite{Lubotzky1994}.  The answer is negative: examples and broad
zero-constant results occur in
\cite{GelanderZuk2002,Osin2002,Hull2016}; a recent finite-quotient
obstruction appears in \cite{Yao2026}.  Lubotzky and Yao subsequently proved
that every infinite finitely generated linear group has zero uniform
Kazhdan constant \cite[Theorem~1.1]{LubotzkyYao2026}.
Uniformity for prescribed families
of generating sets or representations, and uniform nonamenability, have
different quantifiers
\cite{Hadad2007,Pham2022,ArzhantsevaEtAl2005}.  The elementary observation
behind this note is that quasifiniteness converts every infinite subset into
a generating set, so a generating-set-uniform spectral gap becomes a gap at
infinity.  The contribution of this note is the resulting displacement,
action, and coefficient estimates; all construction-theoretic inputs are
quoted from their sources.

\section{Preliminaries}\label{sec:preliminaries}

Throughout, \(G\) is a finitely generated discrete group, \(g\to\infty\)
means that \(g\) leaves every finite subset of \(G\), and complex Hilbert
space inner products are linear in the first variable.

\begin{definition}
An infinite group is \emph{quasifinite} if every proper subgroup is finite.
\end{definition}

For a finite generating set \(S\), put
\begin{align}
 \kappa(G,S)
 &=\inf_{\substack{\pi:G\to\mathcal U(\calH)\\ \calH^G=\{0\}}}
   \ \inf_{\norm{\xi}=1}\ \max_{s\in S}\norm{\pi(s)\xi-\xi},
 \label{eq:kazhdan}\tag{2.1}\\
 \ku(G)&=\inf_S\kappa(G,S),
 \label{eq:uniform-kazhdan}\tag{2.2}
\end{align}
where the last infimum ranges over all finite generating sets, without a
cardinality bound.

The exclusion of invariant vectors in \eqref{eq:kazhdan} is the standard
norm-valued convention \cite{Kazhdan1967,BekkaHarpeValette2008}.\footnote{It
is omitted from the displayed prose in \cite{OsinSonkin2006}, although the
subsequent assertions there require it.}

For an ergodic p.m.p. action \(G\curvearrowright(X,\mu)\), let
\(U_gf=f\circ g^{-1}\) be the Koopman representation on
\(L^2_0(X)=L^2(X)\ominus\C\one\).  It has no invariant vector.  Schmidt's
rigid-set criterion says that the action is mildly mixing precisely when
\[
 \liminf_{g\to\infty}\mu(gA\mathbin\triangle A)>0
 \quad(0<\mu(A)<1),                                      \tag{2.3}
\]
\cite[Proposition~2.3]{Schmidt1984}.  The group has Property~\(M\) when every
ergodic p.m.p. action on a standard nonatomic probability space is mildly
mixing \cite[Definition~3.1]{Schmidt1984}.

\section{The uniform displacement theorem}\label{sec:main}

\begin{theorem}\label{thm:main}
Let \(G\) be infinite, finitely generated, and quasifinite, and suppose that
\(\kappa=\ku(G)>0\).
\begin{enumerate}
\item If \(\pi:G\to\mathcal U(\calH)\) has no invariant vector and
      \(\xi\in\calH\) is a unit vector, then
      \[
        \{g\in G:\norm{\pi(g)\xi-\xi}<\kappa\}
      \]
      is finite.
\item Every ergodic p.m.p. action \(G\curvearrowright(X,\mu)\) and every
      measurable \(A\) with \(0<\mu(A)<1\) satisfy, for all but finitely
      many \(g\),
      \[
        \mu(gA\mathbin\triangle A)
        \geq\kappa^2\mu(A)(1-\mu(A)).                    \tag{3.1}
      \]
      In particular, \(G\) has Property~\(M\).
\item For every such representation \(\pi\), every \(v\ne0\), and all but
      finitely many \(g\),
      \[
        \abs{\inner{\pi(g)v}{v}}
        \leq \sqrt{1-\frac{\kappa^2}{2}}\,\norm v^2.     \tag{3.2}
      \]
\end{enumerate}
\end{theorem}

The passage from displacement to the absolute value in part~(3) uses two
standard facts, included to make the dependence explicit.

\begin{lemma}\label{lem:map}
Every infinite finitely generated quasifinite group is minimally almost
periodic: every finite-dimensional unitary representation is trivial.
\end{lemma}

\begin{proof}
The group is torsion: an infinite cyclic subgroup would be the whole group,
which is impossible because an infinite cyclic group has an infinite proper
subgroup.  The image of a finite-dimensional unitary representation is a
finitely generated periodic complex linear group, hence finite by Schur's
theorem \cite{Schur1911}.  Its kernel is infinite and therefore, by
quasifiniteness, is the whole group.
\end{proof}

\begin{lemma}\label{lem:tensor}
If \(G\) is minimally almost periodic and \(\pi\) has no invariant vector,
then \(\pi\otimes\overline\pi\) has no invariant vector.
\end{lemma}

\begin{proof}
The representation \(\pi\) has no nonzero finite-dimensional invariant
subspace.  Under
\(\calH\otimes\overline{\calH}\cong\HS(\calH)\), an invariant vector would
give a nonzero Hilbert--Schmidt operator \(T\) commuting with \(\pi(G)\).
A nonzero eigenspace of the compact positive operator \(T^*T\) would then be
a nonzero finite-dimensional invariant subspace, a contradiction.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
For part~(1), call the displayed set \(E\) and suppose that it is infinite.
Then \(\langle E\rangle=G\) by quasifiniteness.  Since \(G\) is finitely
generated, finitely many elements \(F\subset E\) generate \(G\).  Hence
\[
 \kappa(G,F)
 \leq\max_{s\in F}\norm{\pi(s)\xi-\xi}
 <\kappa
 \leq\kappa(G,F),
\]
a contradiction.

For part~(2), set \(p=\mu(A)\) and
\(\xi=(\one_A-p\one)/\sqrt{p(1-p)}\in L^2_0(X)\).  Then
\[
 \norm{U_g\xi-\xi}^2
 =\frac{\mu(gA\mathbin\triangle A)}{p(1-p)}.
\]
Part~(1) gives (3.1) outside a finite set.  Its positive lower bound and
(2.3) prove Property~\(M\).

For part~(3), put \(\eta=v/\norm v\),
\(\rho=\pi\otimes\overline\pi\), and
\(\zeta=\eta\otimes\overline\eta\).  Lemmas~\ref{lem:map} and
\ref{lem:tensor} allow part~(1) to be applied to \(\rho,\zeta\); thus, off a
finite set,
\[
 \kappa^2
 \leq\norm{\rho(g)\zeta-\zeta}^2
 =2\bigl(1-\abs{\inner{\pi(g)\eta}{\eta}}^2\bigr).
\]
This is (3.2).  Finally \(\kappa\leq\sqrt2\), since the left regular
representation and \(\delta_e\) give
\(\kappa(G,S)\leq\sqrt2\) for every finite generating set \(S\).  The
radicand is therefore nonnegative and, because \(\kappa>0\), strictly less
than one.
\end{proof}

\section{The Osin--Sonkin construction}\label{sec:osin-sonkin}

We give a corrected proof of the Osin--Sonkin deduction, with all constants
and noncommutation hypotheses made explicit.  There is an important
boundary to the claimed completeness.  The argument below is complete once
the graded-diagram package (GD0)--(GD4) is granted.  Proving that package
from first principles would require reproducing the least-parameter
machinery and the diagram theory of Chapters~8--9 of
\cite{Olshanskii1991Book}, together with Semenov's hyperbolic-source
modifications \cite{SemenovThesis1994,Semenov2017Full}.  The latter is an
unpublished 31-page author manuscript; no single published self-contained
proof of the combined quantitative package appears to be available.
Semenov's short
published announcement states the basic quotient theorem
\cite{Semenov1993}.  Kulikova later invokes thesis Theorem~2.6, identified
there with Semenov's 1993 Theorem~1
\cite[Theorem~4, Section~4]{Kulikova2023}; this is later expert use, not an
independent proof of the input.  Neither published paper supplies the
quantitative pair estimates needed for Property~\(U\).

For a subset \(S\) of a group, \(\abs{g}_S\) denotes word length with
respect to \(S\cup S^{-1}\).  We use
\([x,y]=xyx^{-1}y^{-1}\).

\begin{definition}\label{def:property-u}
Let \(Q\) be generated by a finite set \(X\).  The group \(Q\) has
\emph{Property~\(U\)} if there is \(C_U<\infty\) such that, for every
generating set \(Y\) of \(Q\), there is \(t\in Q\) satisfying
\[
 \max_{x\in X}\abs{t^{-1}xt}_Y\leq C_U.
\]
\end{definition}

\subsection{The graded-diagram input}\label{subsec:gd-input}

Let
\[
 \Gamma=\langle a_1,\ldots,a_m\mid\mathcal R_0\rangle
\]
be a non-elementary torsion-free hyperbolic group.  We take
\(\mathcal R_0\) to be the complete set of cyclically reduced
\(X\)-words trivial in \(\Gamma\), after deleting trivial generators
and inverse or duplicate generators, and put
\(X=\{a_1,\ldots,a_m\}\).  The Ol'shanskii--Semenov least-parameter
construction supplies constants
\[
 d>0,\qquad h,n,N\in\mathbb N,\qquad \sigma>0,
 \qquad N\ \text{odd},\qquad \sigma<N/4,
\]
a graded quotient \(Q=G(\infty)\) of \(\Gamma\), and sets of periods.
For an \(X\)-word \(W\), write \(\wlen{W}\) for its combinatorial
length.  Capital letters denoting periods or auxiliary words always
denote chosen \(X\)-words as well as the elements they represent; equations
between them are equations in the indicated group.  The exact input used
below is:

\begin{enumerate}
\item[(GD0)] If \(A\) is a period of rank \(i\), the rank-\(i\) relations
contain \(A^N=1\).  For each \(A\), a maximal set \(\mathcal G_A\) is chosen
of words \(T\) satisfying
\[
 0<\wlen{T}<d\wlen{A},
\]
with at most one minimum-length representative from each double coset
\(\langle A\rangle T\langle A\rangle\) in \(G(i-1)\).  If
\[
 a_j\notin\langle A\rangle,\qquad
 a_j\notin\langle A\rangle a_k\langle A\rangle\quad(k<j),
\]
then for every
\(T\in\mathcal G_A\setminus\langle A\rangle a_j\langle A\rangle\)
the rank-\(i\) relations contain
\begin{equation}
 a_jA^{n+j-1}T A^{n+m+j-1}\cdots
 T A^{n+m(h-1)+j-1}=1.                    \tag{GD0.1}
\end{equation}
There are \(h\) displayed powers of \(A\) and \(h-1\) occurrences of \(T\).

\item[(GD1)] The group \(Q\) is infinite, has exponent dividing \(N\), and
every proper subgroup of \(Q\) is cyclic of order dividing \(N\).

\item[(GD2)] Every period has order exactly \(N\) in \(Q\).  Every
nonidentity element is conjugate to a nonzero power of a period, and
\(C_Q(x)\) is cyclic of order exactly \(N\) for every \(x\ne1\).

\item[(GD3)] Let \(C\) be a period, let
\(\sigma<k<N/2\), and let \(W\) have minimum \(\wlen{\cdot}\)-length in
\(\langle C^k\rangle W\langle C^k\rangle\).  Suppose that
\([C^k,W]\ne1\) and that \([C^k,W]\) is conjugate in \(Q\) to
\(A^\ell\), where \(A\) is a period and
\(1\leq\abs{\ell}\leq (N-1)/2\).  Then \(\abs{\ell}\leq\sigma\),
and there exist \(g\in Q\) and an \(X\)-word \(B\) such that
\[
 [C^k,W]=g^{-1}A^\ell g,\qquad
 B=gC^kg^{-1}\ \text{in }Q,\qquad
 \wlen{B}<d\wlen{A}.
\]

\item[(GD4)] Let \(A\) be a period, let \(B\) be a chosen \(X\)-word,
and suppose
\[
 0<\abs{\ell}\leq\sigma,\qquad
 [A^\ell,B]\ne1,\qquad
 \wlen{B}<d\wlen{A}.
\]
Then there are \(s\in\mathbb Z\), \(z\in Q\),
\(\varepsilon\in\{-1,1\}\), a period \(F\), and a word \(T\) such that
\[
 F^\varepsilon=z^{-1}BA^{\ell s}z,\qquad T=z^{-1}Bz,\qquad
 [F^\varepsilon,T]\ne1,\qquad \wlen{T}<d\wlen{F}.
\]
\end{enumerate}

The exact source map is as follows.  Item (GD0) is Semenov's
Section~11, and condition \(R\) is his Lemma~11.2 (27.2).
Item (GD1) is Theorem~A, using Theorems~10.1, 10.2, 10.5 and
Lemma~11.4 (27.4); item (GD2) uses Theorems~10.2, 10.4 and 10.5
\cite[pp.~29--31]{Semenov2017Full}.  Item (GD3) is Semenov's
Lemma~9.23 (25.21) \cite[p.~27]{Semenov2017Full}, modifying
Ol'shanskii's Lemma~25.21
\cite[pp.~281--282]{Olshanskii1991Book}.  Item (GD4) is the
intermediate pair-straightening step in the last five paragraphs of
Ol'shanskii's Lemma~27.3 \cite[pp.~301--303]{Olshanskii1991Book}, with
the hyperbolic-source changes specified in Semenov's Lemma~11.3
\cite[pp.~30--31]{Semenov2017Full}.  Semenov also uses the published
hyperbolic estimates
\cite[Lemmas~26, 27, 34, 42, pp.~527--528, 531--532, 538]{Olshanskii1992};
that paper refers its Sections~6--9 back to the book's definitions and
proofs \cite[p.~520]{Olshanskii1992}.

The noncommutation needed to invoke (GD4) is asserted immediately
after Lemma~25.21 is applied inside the source proof of Lemma~27.3.
Rather than importing it, we derive it from (GD1)--(GD2) in
Lemma~\ref{lem:pair-reduction}.  For comparison,
\cite[Lemmas~2.4--2.5, pp.~406--407]{Osin2007} gives the parallel
free-Burnside-group argument, not the hyperbolic-source modification;
its displayed Lemma~2.5 has the same local zero-exponent omission.
The literal formulation of \cite[Lemma~2.5]{OsinSonkin2006} allows
\(\ell=0\), in which case it is false; (GD4) records the corrected
hypothesis used in the application.

Put
\[
 q=\frac{N-1}{2},\qquad I=\lfloor\sigma\rfloor+1,\qquad
 E=2N+2I,\qquad M=(q+1)E,
\]
and
\begin{equation}
 L=\max\{N,\;hq+(h-1)N+2q\},\qquad C_U=LM. \tag{4.1}
\end{equation}
These constants depend only on the fixed quotient and its fixed
generating set.

\begin{lemma}[Normalizer rigidity]\label{lem:normalizer-rigidity}
If \(1\ne x\in Q\) and \(yxy^{-1}\) commutes with \(x\), then \(y\)
commutes with \(x\).
\end{lemma}

\begin{proof}
Let \(H=C_Q(x)\).  By (GD2), \(H\) is cyclic of order \(N\).  If
\(a=yxy^{-1}\) commutes with \(x\), then \(a\in H\).  Since \(H\) is
abelian, \(H\subseteq C_Q(a)\); both groups have order \(N\), so
\[
 yHy^{-1}=C_Q(yxy^{-1})=C_Q(a)=H.
\]
If \(\langle H,y\rangle\) is cyclic, then \(y\) commutes with \(x\).
Otherwise (GD1) forces \(\langle H,y\rangle=Q\), and \(H\) is a
nontrivial finite normal subgroup.  The conjugation homomorphism
\(Q\to\operatorname{Aut}(H)\) has an infinite kernel, which by (GD1)
must be all of \(Q\).  Then \(H\subseteq Z(Q)\), contradicting
\(C_Q(x)=H\ne Q\).
\end{proof}

\begin{lemma}[A short period pair]\label{lem:short-pair}
Let \(F\) be a period and let \(T\) be a word satisfying
\[
 [F,T]\ne1,\qquad \wlen{T}<d\wlen{F}.
\]
Then
\[
 \max_{1\leq j\leq m}\abs{a_j}_{\{F,T\}}\leq L.
\]
\end{lemma}

\begin{proof}
Let \(i\) be the rank of \(F\), and make the following double-coset
comparisons in \(G(i-1)\).  Since \(F^N=1\) in \(Q\), every power of
\(F\) has \(\{F,T\}\)-length at most \(q\).

Fix \(j\).  If \(a_j\in\langle F\rangle\), then
\(\abs{a_j}_{\{F,T\}}\le q\).  If
\(T\in\langle F\rangle a_j\langle F\rangle\), reducing the two
flanking powers modulo \(N\) gives
\[
 \abs{a_j}_{\{F,T\}}\leq 2q+1=N.           \tag{4.2}
\]

In the remaining case, in \(G(i-1)\),
\[
 a_j\notin\langle F\rangle,\qquad
 T\notin\langle F\rangle a_j\langle F\rangle.
\]
Let \(j_0\) be the least index such that
\(a_{j_0}\in\langle F\rangle a_j\langle F\rangle\).  Thus
\(a_{j_0}\notin\langle F\rangle\), and by minimality it lies in no
double coset \(\langle F\rangle a_k\langle F\rangle\) with \(k<j_0\).
The double coset
\(\langle F\rangle T\langle F\rangle\) has a representative
\(T_1\in\mathcal G_F\): the word \(T\) itself satisfies the required
length bound, this double coset is nontrivial because \([F,T]\ne1\),
and maximality supplies a minimum-length representative.
Moreover,
\(T_1\notin\langle F\rangle a_{j_0}\langle F\rangle\).  Thus (GD0)
contains the relation
\begin{equation}
 a_{j_0}F^{n+j_0-1}T_1F^{n+m+j_0-1}\cdots
 T_1F^{n+m(h-1)+j_0-1}=1.                \tag{4.3}
\end{equation}
The index in (4.3) is \(j_0\), rather than the \(j\) printed at the
corresponding point of \cite{OsinSonkin2006}.

Because \(T_1\in\langle F\rangle T\langle F\rangle\),
\[
 \abs{T_1}_{\{F,T\}}\leq2q+1=N.            \tag{4.4}
\]
Solving (4.3) for \(a_{j_0}\) gives
\[
 \abs{a_{j_0}}_{\{F,T\}}\leq hq+(h-1)N.
\]
Finally \(a_j\in\langle F\rangle a_{j_0}\langle F\rangle\), whence
\[
 \abs{a_j}_{\{F,T\}}\leq hq+(h-1)N+2q\leq L.
\]
Together with (4.2), this proves the assertion.  In particular, the
safe constant depends on \(h\), and the bound in (4.4) need not be
strict.
\end{proof}

\begin{lemma}[Uniform pair reduction]\label{lem:pair-reduction}
For every \(u,v\in Q\) with \([u,v]\ne1\), there are a period \(F\), a
 word \(T\), and \(t\in Q\) such that
\[
 [F,T]\ne1,\qquad \wlen{T}<d\wlen{F},
\]
and
\begin{equation}
 \max\{\abs{t^{-1}Ft}_{\{u,v\}},
          \abs{t^{-1}Tt}_{\{u,v\}}\}\leq M.             \tag{4.5}
\end{equation}
\end{lemma}

\begin{proof}
By (GD2),
\[
 u=P^{-1}C^{k'}P
\]
for a period \(C\).  Replace the pair \((u,v)\) by
\((u^{-1},v)\) and relabel it, if necessary; the symmetric word
metric is unchanged and noncommutation is preserved.  We may then
assume \(1\leq k'\leq q\).  If \(k'>\sigma\), put \(i=1\);
otherwise put
\[
 i=\left\lfloor\frac{\sigma}{k'}\right\rfloor+1.
\]
Then \(i\leq I\) and
\[
 \sigma<k:=ik'<N/2.                       \tag{4.6}
\]
Indeed, if \(k'>\sigma\), then \(k=k'\leq q<N/2\); otherwise
\(k\leq\sigma+k'\leq2\sigma<N/2\).
The exact order assertion in (GD2) gives \(C^k\ne1\).  If \(u^i\)
commuted with \(v\), then \(u,v\in C_Q(u^i)\), a cyclic group,
contrary to \([u,v]\ne1\).  Therefore
\[
 [C^k,PvP^{-1}]=P[u^i,v]P^{-1}\ne1.        \tag{4.7}
\]

Put \(x=C^k\), \(W_0=PvP^{-1}\), and choose \(W\) of minimum
\(\wlen{\cdot}\)-length in
\(\langle x\rangle W_0\langle x\rangle\).  There are
\(\alpha,\beta\in\mathbb Z\), with
\(\abs{\alpha},\abs{\beta}\leq q\), such that
\[
 W=x^\alpha W_0x^\beta.
\]
The element \(W\) does not commute with \(x\), since
\[
 [x,x^\alpha W_0x^\beta]=1
 \quad\Longleftrightarrow\quad [x,W_0]=1.
\]
Moreover
\[
 P^{-1}WP=u^{i\alpha}vu^{i\beta}.
\]
Reducing both \(u\)-exponents modulo \(N\), rather than using the
incorrect signed estimate printed in \cite{OsinSonkin2006}, gives
\begin{equation}
 \abs{P^{-1}WP}_{\{u,v\}}\leq2q+1=N.       \tag{4.8}
\end{equation}

Let \(D=[x,W]\).  From (4.6)--(4.8),
\begin{equation}
 \abs{P^{-1}xP}_{\{u,v\}}\leq I,\qquad
 \abs{P^{-1}DP}_{\{u,v\}}\leq2I+2N=E.      \tag{4.9}
\end{equation}
By (GD2), choose a period \(A\) and an integer \(\ell\) with
\(1\leq\abs{\ell}\leq q\) such that \(D\) is conjugate to \(A^\ell\).
By (GD3), \(\abs{\ell}\leq\sigma\), and there exist \(g\in Q\) and
an \(X\)-word \(B\) such that
\[
 D=g^{-1}A^\ell g,\qquad
 B=gxg^{-1}\ \text{in }Q,\qquad
 \wlen{B}<d\wlen{A}.                       \tag{4.10}
\]

For clarity, we verify the noncommutation that is tacit in the
preprint.  If \(D\) commuted with \(x\), put \(a=WxW^{-1}\).
Since \(D=xa^{-1}\), the elements \(a\) and \(x\) would commute.
Lemma~\ref{lem:normalizer-rigidity}, applied with \(y=W\), would make
\(W\) commute with \(x\), contradicting the equivalence above and
(4.7).  Thus
\([D,x]\ne1\), and simultaneous conjugation by \(g\) gives
\[
 [A^\ell,B]\ne1.                           \tag{4.11}
\]

Apply (GD4), obtaining \(s,z,\varepsilon,F,T\).  Since
\([F^\varepsilon,T]\ne1\), also \([F,T]\ne1\).  Replace \(s\) by its
residue in \([-q,q]\); this does not change \(BA^{\ell s}\).  Since
\[
 g^{-1}Bg=x,\qquad g^{-1}A^\ell g=D,
\]
we have
\[
 g^{-1}BA^{\ell s}g=xD^s,\qquad g^{-1}Bg=x.
\]
Set \(t=z^{-1}gP\).  Equations (4.9) and the preceding identities give
\[
 \abs{t^{-1}Ft}_{\{u,v\}}
 =\abs{t^{-1}F^\varepsilon t}_{\{u,v\}}
 \leq I+qE\leq(q+1)E=M,\qquad
 \abs{t^{-1}Tt}_{\{u,v\}}\leq I\leq M.
\]
\end{proof}

\begin{theorem}[Corrected Osin--Sonkin deduction]\label{thm:osin-sonkin-u}
Assume (GD0)--(GD4).  Then \(Q\) has Property~\(U\) with respect to
\(X\).  More precisely, every generating set \(Y\) of \(Q\) admits
\(t\in Q\) such that
\[
 \max_{1\leq j\leq m}\abs{t^{-1}a_jt}_Y\leq C_U.
\]
\end{theorem}

\begin{proof}
The group \(Q\) is nonabelian by (GD2), so \(Y\) contains
noncommuting elements \(u,v\).  The subgroup \(\langle u,v\rangle\)
is noncyclic and therefore equals \(Q\) by (GD1).  In particular,
\(\abs{g}_Y\leq\abs{g}_{\{u,v\}}\).

Apply Lemma~\ref{lem:pair-reduction} to obtain \(F,T,t\).  By
Lemma~\ref{lem:short-pair}, each \(a_j\) is represented by a word of
length at most \(L\) in \(F^{\pm1},T^{\pm1}\).  Conjugating that word
by \(t\) and using (4.5) gives
\[
 \abs{t^{-1}a_jt}_Y
 \leq\abs{t^{-1}a_jt}_{\{u,v\}}\leq LM=C_U.
\]
\end{proof}

\begin{lemma}[Kazhdan comparison]\label{lem:kazhdan-comparison}
Let \(S,T\) be finite generating sets of a group \(G\), and let
\(g\in G\).  Then
\[
 \kappa(G,g^{-1}Sg)=\kappa(G,S).
\]
If \(D=\max_{s\in S}\abs{s}_T\), then
\[
 \kappa(G,T)\geq D^{-1}\kappa(G,S).
\]
\end{lemma}

\begin{proof}
For a unitary representation \(\pi\) and a unit vector \(\xi\),
\[
 \norm{\pi(g^{-1}sg)\xi-\xi}
 =\norm{\pi(s)\eta-\eta},\qquad \eta=\pi(g)\xi.
\]
The map \(\xi\mapsto\eta\) permutes the unit sphere, proving
conjugacy invariance after taking the two infima in
\eqref{eq:kazhdan}.  This corrects the false pointwise equality in
\cite[Lemma~3.1]{OsinSonkin2006}.

If \(s=t_1\cdots t_r\), where \(r\leq D\) and
\(t_i\in T\cup T^{-1}\), telescoping gives
\[
 \norm{\pi(s)\xi-\xi}
 \leq r\max_{t\in T}\norm{\pi(t)\xi-\xi}.
\]
Here
\(\norm{\pi(t^{-1})\xi-\xi}=\norm{\pi(t)\xi-\xi}\), so inverse
letters are controlled by the same maximum.
Taking maxima and infima gives the second assertion.
\end{proof}

\begin{corollary}\label{cor:uniform-kazhdan-os}
If the initial group \(\Gamma\) has Property~\((T)\), then
\[
 \ku(Q)\geq C_U^{-1}\kappa(Q,X)>0.
\]
\end{corollary}

\begin{proof}
Property~\((T)\) passes to quotients, so \(\kappa(Q,X)>0\).  For any
finite generating set \(Y\), Theorem~\ref{thm:osin-sonkin-u} supplies
\(t\) with
\(\max_{x\in X}\abs{t^{-1}xt}_Y\leq C_U\).
Lemma~\ref{lem:kazhdan-comparison} yields
\[
 \kappa(Q,Y)\geq
 C_U^{-1}\kappa(Q,t^{-1}Xt)
 =C_U^{-1}\kappa(Q,X).
\]
Taking the infimum over \(Y\) proves the result.
\end{proof}

\begin{corollary}[Conditional on (GD0)--(GD4)]
\label{cor:osin-sonkin}
The quotient \(Q=G(\infty)\) constructed from a torsion-free
non-elementary hyperbolic Kazhdan group has Property~\(M\) and satisfies
(3.1)--(3.2).  It is an infinite finitely generated simple ICC group of
bounded odd exponent, and every proper subgroup is finite cyclic.
\end{corollary}

\begin{proof}
The group is finitely generated because it is a quotient of \(\Gamma\).
By (GD1) and Corollary~\ref{cor:uniform-kazhdan-os}, it is infinite,
quasifinite, and has positive uniform Kazhdan constant, so
Theorem~\ref{thm:main} applies.

By (GD2), every nontrivial centralizer is finite, so every nontrivial
conjugacy class is infinite and \(Q\) is ICC.  A nontrivial proper
normal subgroup would be finite by quasifiniteness, contradicting ICC;
hence \(Q\) is simple.  Finally, (GD1) gives the stated cyclic proper
subgroups and bounded odd exponent.
\end{proof}

The algebraic and uniform Kazhdan properties are the Osin--Sonkin
claims reconstructed above.  The new consequences are Property~\(M\)
and the quantitative estimates.  The conditional label records exactly
the remaining source-verification issue: the elementary reduction is
now complete, while the hyperbolic-source proof of (GD0)--(GD4) remains
the composite Ol'shanskii--Semenov input described at the start of this
section.

\section{Concluding remarks and questions}\label{sec:conclusion}

Fix a Kazhdan hyperbolic source in Ozawa's construction.  The resulting
groups \(\Gamma_\alpha\) are finitely generated Kazhdan quotients,
quasifinite and ICC, and therefore simple \cite{Ozawa2004}.  Ozawa proves
quasifiniteness, but does not assert bounded exponent, prescribed cyclic
proper subgroups, Property~\(U\), or a positive uniform Kazhdan constant.
Theorem~\ref{thm:main} therefore suggests: \emph{Do Ozawa's groups
\(\Gamma_\alpha\) have Schmidt's Property~\(M\)?  Does
\(\ku(\Gamma_\alpha)>0\) hold for any, or for every, member of the family?}
The second question gives a sufficient route to the first, not a necessary
condition.  Ordinary Property~\((T)\) supplies a positive constant for each
finite generating set separately; it gives no common lower bound for the
tail-dependent generating sets arising from a sequence escaping to
infinity.

Ozawa's motivation was operator algebraic.  His Theorem~2 says that, for a
fixed separable \(\mathrm{II}_1\)-factor \(M\), the unitary group
\(\mathcal U(M)\) contains an isomorphic copy of \(\Gamma_\alpha\) for at
most countably many \(\alpha\).
His Corollary~3 says that \(M\) contains a subfactor stably equivalent to
\(L(\Gamma_\alpha)\) for at most countably many \(\alpha\)
\cite{Ozawa2004}.  Neither statement asserts that the factors
\(L(\Gamma_\alpha)\) are pairwise nonisomorphic.

The broader landscape changed in August 2026.  Connes had conjectured that
an ICC Kazhdan group is determined by its group von Neumann algebra
\cite[p.~45]{Connes1982}.  An OpenAI manuscript constructs a countably
infinite family of pairwise nonisomorphic, mutually commensurable, finitely
generated ICC Kazhdan groups with mutually isomorphic group factors
\cite[Chapter~4]{OpenAI2026TenAdvances}.  A contemporaneous preprint of Zhou
constructs an explicit pair of nonisomorphic ICC Kazhdan groups with
isomorphic group factors \cite{Zhou2026ICC}.  These very recent sources
refute the original conjecture for arbitrary ICC Kazhdan groups, not the
narrower higher-rank-lattice variant discussed in
\cite[Section~1]{Zhou2026ICC}.

Neither construction enters the quasifinite regime.  The OpenAI groups and
Zhou's groups are semidirect products with infinite proper abelian normal
kernels.  Schmidt's subgroup obstruction
\cite[Lemma~6.2]{Schmidt1984} therefore shows that none has Property~\(M\).
These examples do not determine whether Property~\(M\) is invariant under
isomorphism of group factors, and they do not resolve the questions above.
Finally, our coefficient bound is a gap from modulus one, not decay to zero,
so it proves \(M\) rather than \(S\).  Since an infinite amenable Kazhdan
group cannot exist, the method does not address the amenable quasifinite
case.

\section*{Acknowledgments}

The research prompt was adapted from Chao Xu's
\emph{AI Agents for the Working Mathematician}~\cite{xu2026agents}.

OpenAI's GPT-5.6 Sol, used with ultra reasoning effort, assisted substantially
in drafting this manuscript.  Its role included organizing the literature
review, formulating and revising the exposition, checking internal
consistency, and preparing the \TeX{} source.  An OpenAI assistant also helped prepare
this review copy and check its bibliography and exposition.  The mathematical claims,
source attributions, and any remaining errors are the author's
responsibility.

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