\documentclass[a4paper,11pt]{amsproc} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[a4paper,margin=25mm,headheight=14pt,footskip=10mm]{geometry} \usepackage{fancyhdr} \usepackage{amssymb,mathtools} \usepackage{microtype} \usepackage{xcolor} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,calc} % Use amsart theorem and proof formatting within amsproc. % Empty spacing arguments retain the AMS defaults; empty indent means flush left. \newtheoremstyle{plain}{}{}{\itshape}{}{\bfseries}{.}{5pt plus 1pt minus 1pt}{} \newtheoremstyle{definition}{}{}{\normalfont}{}{\bfseries}{.}{5pt plus 1pt minus 1pt}{} \newtheoremstyle{remark}{}{}{\normalfont}{}{\itshape}{.}{5pt plus 1pt minus 1pt}{} % The amsart proof environment, including its QED handling. \makeatletter \renewenvironment{proof}[1][\proofname]{\par \pushQED{\qed}% \normalfont \topsep6\p@\@plus6\p@\relax \trivlist \item[\hskip\labelsep\itshape #1\@addpunct{.}]\ignorespaces }{% \popQED\endtrivlist\@endpefalse } \makeatother \usepackage[backref=section]{hyperref} \usepackage{bookmark} \usepackage[alphabetic]{amsrefs} \renewcommand*{\biblistfont}{\normalfont\small} \renewcommand*{\PrintDOI}[1]{\href{https://doi.org/#1}{\nolinkurl{doi:#1}}} \urlstyle{same} \definecolor{citegreen}{rgb}{0,0.5,0.15} \definecolor{urlgreen}{rgb}{0,0.35,0.15} \definecolor{linkblue}{rgb}{0,0.4,1} \definecolor{citebordergreen}{rgb}{0,0.9,0.37} \hypersetup{colorlinks=true, linkbordercolor={0 0 1}, linkcolor=blue, urlcolor=urlgreen, unicode=true, citecolor=citegreen, linkbordercolor={0 0.4 1}, citebordercolor={0 0.9 0.37}, pdftitle={A geometric account of the solution to the Dixmier unitarisability problem}} % Record the enclosing subsection or section, not the latest theorem/equation. \makeatletter \let\originalbackout\Hy@backout \renewcommand*{\Hy@backout}[1]{% \begingroup \ifnum\value{subsection}>0 \edef\@currentlabel{\thesubsection}% \edef\@currentHref{subsection.\theHsubsection}% \else\ifnum\value{section}>0 \edef\@currentlabel{\thesection}% \edef\@currentHref{section.\theHsection}% \else \def\@currentlabel{0}% \def\@currentHref{backref-introduction}% \fi\fi \originalbackout{#1}% \endgroup } \makeatother % Keep the compact blue back-references when amsrefs prints each entry. \makeatletter \AtBeginDocument{% \renewcommand*{\PrintBackRefs}[1]{% \@ifundefined{br@#1}{}{% \begingroup \hypersetup{linkcolor=blue}\color{blue}% \space\ensuremath{\uparrow}\,\csname br@#1\endcsname \endgroup }% }% } \makeatother \fancyhf{} \fancyhead[C]{\normalfont\footnotesize\papertitle} \fancyhead[LE,RO]{\normalfont\small\thepage} \renewcommand{\headrulewidth}{0.4pt} \pagestyle{fancy} \setlength{\emergencystretch}{2em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem*{maintheorem}{Main theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newcommand{\C}{\mathbb C} \newcommand{\R}{\mathbb R} \newcommand{\N}{\mathbb N} \newcommand{\E}{\mathbb E} \newcommand{\Prob}{\mathbb P} \newcommand{\one}{\mathbf 1} \newcommand{\LG}{LG} \newcommand{\RG}{RG} \newcommand{\cB}{\mathcal B} \newcommand{\cU}{\mathcal U} \newcommand{\cW}{\mathcal W} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\row}{row} \DeclareMathOperator{\col}{col} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\rank}{rank} \makeatletter \def\@adminfootnotes{% \let\@makefnmark\relax \let\@thefnmark\relax \ifx\@empty\@date\else \@footnotetext{\@setdate}\fi%% <------ added \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi \ifx\@empty\thankses\else \@footnotetext{% \def\par{\let\par\@par}\@setthanks}% \fi } \makeatother \linespread{1.2} \newcommand{\papertitle}{A geometric account of the solution to the Dixmier unitarisability problem} \title{\papertitle} \author{Vadim Alekseev} \date{October 8, 2026, last revised \today} \begin{document} \begin{abstract} This note presents a more geometric account of OpenAI's solution to the Dixmier unitarisability problem. Some parts of the original proof are reinterpreted using the existence of a positive divergence flow on a non-amenable group, and the randomized construction for operator norm control of the Markov operator with matrix coefficients is replaced by one coming from free groups. \end{abstract} \maketitle\thispagestyle{empty} \section*{Introduction} \hypertarget{backref-introduction}{} A representation $\pi:G\to\mathrm{GL}(H)$ on a complex Hilbert space is \emph{uniformly bounded} if $|\pi|:=\sup_{g\in G}\norm{\pi(g)}<\infty$, and \emph{unitarisable} if a bounded invertible operator $V$ makes $V\pi(g)V^{-1}$ unitary for every $g\in G$. A group is unitarisable if all its uniformly bounded Hilbert space representations are unitarisable. Sz.-Nagy's theorem on operators with uniformly bounded integer powers established the case $G=\mathbb Z$ \cite{sznagy}. Day and Dixmier extended the invariant-mean argument to amenable groups \cites{day,dixmier}; see also Nakamura--Takeda \cite{nt}. Dixmier immediately asked about the converse \cite{dixmier}*{\S\,5}. OpenAI's preprint \cite{openai} proves this converse for discrete groups. The purpose of this note is to give a geometric account of its operator obstruction and a construction of the required positive matrix coefficients using free products of finite groups. We work throughout with countable discrete groups (since the reduction to the general case is straightforward) and prove the following statement. Together with the Day--Dixmier theorem~\cites{day,dixmier}, this characterizes amenability among discrete groups. \begin{maintheorem} Every countable nonamenable discrete group is nonunitarisable. \end{maintheorem} The history of the problem involves several different ways of detecting nonunitarisability. Ehrenpreis and Mautner produced the first examples for $\mathrm{SL}_2(\R)$ in 1955 \cite{ehrenpreis}, with a detailed development by Kunze and Stein \cite{ks}. Explicit constructions for nonabelian free groups were obtained subsequently, including the analytic families of Pytlik and Szwarc \cite{ps}. Since unitarisability passes to subgroups, these results imply nonunitarisability for every group containing a nonabelian free subgroup. The remaining difficulty was to obtain an obstruction for general nonamenable groups, including those without such subgroups. Later work developed quantitative criteria for unitarisability \cites{pisier98,pisier05}. Random-forest methods and Osin's constructions produced nonunitarisable groups without free subgroups \cites{em,osin}, while measurable-group techniques yielded results for wreath products, later extended to finite lamp groups by Alpeev \cites{monod,gl,mo,alpeev}. Further work connected unitarisability with fixed price and approximation properties, including for groups acting on finite-dimensional CAT$(0)$ cube complexes \cites{thom-forests,vergara}. The analytic approaches closest to the present proof are Littlewood-function methods for amenability and unitarisability \cites{wysoczanski,bf,em}, later related to asymptotic Cheeger constants by Gerasimova, Gruber, Monod, and Thom \cite{ggmt}. The general idea of the proof given here can be explained as follows. Since the group is non-amenable, it has a flow of positive divergence which splits the random-walk operator on the Cayley graph into contributions from walks with positive and non-positive total flow. Thus we can decompose the Markov operators (acting from the right) on bigger and bigger powers of the generating set into the ``non-positive flow'' and ``positive flow'' parts, called $T$ resp. $Q$ (we omit the index counting the powers of the generating set here). Positive divergence makes the relevant non-positive part sparse, which allows to uniformly control the operator norm displacement of $T$ under the $G$-action. At the same time, the resulting tracial $2$-norms of the Markov operators, if one allows suitable matrix amplifications, tend to infinity, which together with sparsity control on $T$ shows via a tracial distance estimate that $T$ move further and further from the right group von Neumann algebra. Therefore the cocycle given at each stage by $T - U(g)TU(g)^{-1}$ remains bounded in the direct sum, but becomes non-inner there, which yields a nonunitarisable uniformly bounded representation. Since this cocycle arises as a sum of inner ones, this also proves that the group is not deformation rigid in the sense of \cite{bot}. \subsection*{AI assistance} AI tools contributed to several stages of the work leading to this paper, starting with OpenAI's solution to the Dixmier unitarisability problem. The present exposition was developed through interaction of GPT-6 Astra with subsequent editing by the author. The author supplied ideas reinterpreting the original proof in a geometric way as well as placing the relevant quantities in the proper operator-algebraic context, which were then made precise by Astra; the usage of group-theoretic sources instead of random matrix methods was suggested by the author and worked out by Astra in detail with subsequent manual editing by the author to improve presentation. The author had written the relevant explanations of the proof ideas and steps and takes responsibility for the mathematical claims, references, and final presentation. \subsection*{Acknowledgments} The author thanks Alon Dogon for pointing out that OpenAI's construction proves that ever non-amenable group is not deformation rigid as well as Martin Finn-Sell, Maria Gerasimova, Narutaka Ozawa and Andreas Thom for valuable comments. \section{Preliminaries and notation} \noindent\textbf{Operator algebras.} Let $G$ be a countable discrete group, with identity $e$. Inner products are linear in the second variable. On $\ell^2(G)$ let \begin{equation*} (\lambda(g)\xi)(x)=\xi(g^{-1}x),\qquad (\rho(g)\xi)(x)=\xi(xg),\qquad LG=\lambda(G)'',\quad RG=\rho(G)''=LG'. \end{equation*} The canonical traces on $LG$ and $RG$, the normalized matrix traces, and their tensor products are denoted by $\tau$: \begin{equation*} \tau(X)=\langle\delta_e,X\delta_e\rangle,\qquad \tau(B)=\frac1k\Tr(B),\qquad \norm{A}_2=\tau(A^*A)^{1/2}. \end{equation*} Every element of $LG\,\overline\otimes\,M_k(\C)$, respectively $RG\,\overline\otimes\,M_k(\C)$, has a unique Fourier expansion converging in $L^2$: \begin{equation}\label{eq:fourier} A=\sum_{g\in G}\lambda(g)\otimes A_g \quad\text{or}\quad A=\sum_{g\in G}\rho(g)\otimes A_g, \qquad \norm{A}_2^2=\sum_g\norm{A_g}_2^2. \end{equation} Put $H_k=\ell^2(G)\otimes\C^k$. Viewing $\delta_x$ as a map $\C\to\ell^2(G)$, define for $X\in\cB(H_k)$ \begin{equation*} \row_x(X)=(\delta_x^*\otimes \one_k)X,\qquad \col_x(X)=X(\delta_x\otimes \one_k). \end{equation*} For $A\in RG\,\overline\otimes\,M_k(\C)$, the trace identities give \begin{equation}\label{eq:trace-row-column} \norm{A}_2^2 =\tau\bigl(\row_e(A)\row_e(A)^*\bigr) =\tau\bigl(\col_e(A)^*\col_e(A)\bigr). \end{equation} Hence $\norm{A}_2\le\norm{\row_e(A)}$ and $\norm{A}_2\le\norm{\col_e(A)}$. \noindent\textbf{Littlewood functions.} Recall that $f\in T^1(G)$ if $f(x^{-1}y)=u(x,y)+v(x,y)$ with \begin{equation*} \norm{f}_{T^1} =\inf_{f(x^{-1}y)=u+v}\bigl(\alpha(u)+\gamma(v)\bigr)<\infty, \quad \alpha(u)=\sup_x\sum_y|u(x,y)|,\quad \gamma(v)=\sup_y\sum_x|v(x,y)|. \end{equation*} See \cite{ggmt}*{Definition 2.3 and Propositions 2.5--2.6}. For a finitely supported $f$ and a decomposition $\rho(f)=T+Q$ with kernels $u,v$, Schur's test gives \begin{equation*} \sup_g\norm{T-\lambda(g)T\lambda(g)^*} \le2\sqrt{\alpha(u)\gamma(v)}. \end{equation*} Indeed the difference has row bound $2\alpha(u)$ and, since it also equals $\lambda(g)Q\lambda(g)^*-Q$, column bound $2\gamma(v)$. Lemma~\ref{lem:walk-operator} gives the corresponding estimate for positive matrix coefficients. %For unitarisable $G$, the standard Littlewood estimate is % $\norm{f}_{\ell^2(G)}\le C_G\norm{f}_{T^1(G)}$; see Bo\.{z}ejko--Fendler \cite{bf} and Epstein--Monod \cite{em}*{Proposition 2.4}. %Here $\norm{f}_{\ell^2(G)}=\norm{\rho(f)}_2$. \section{Positive and non-positive walks} Let $G$ be a countable nonamenable discrete group. By the F\o lner criterion, there is a finite symmetric set $S\subset G\setminus\{e\}$ such that, with $d=|S|$, \begin{equation*} h:=\inf_{\varnothing\ne F\subset G\text{ finite}} \frac{|\partial_E F|}{d|F|}>0, \qquad \partial_EF=\{(x,xs):x\in F,\ s\in S,\ xs\notin F\}. \end{equation*} Fix $0<\eta<\min(h,1)$ and set $\beta=\sqrt{1-\eta^2}$. On the right Cayley graph, an antisymmetric flow gives a signed value to each oriented edge. Its integral along a walk changes sign under reversal. Positive divergence means that a random walk has positive expected flow at every step. Consequently, non-positive walks starting at a fixed vertex are rare; by reversal, positive walks ending at a fixed vertex are also rare. These are the two estimates that will control the two parts of the Markov operator. \begin{lemma}[see {\cite{blps}*{Proposition 2.1}}]\label{lem:flow} There is a real function $b$ on oriented Cayley edges satisfying \begin{equation*} b(x,y)=-b(y,x),\qquad |b(x,y)|\le1,\qquad \frac1d\sum_{s\in S}b(x,xs)=\eta\quad(x\in G). \end{equation*} \end{lemma} \begin{proof} For a finite set $K\subset G$, form a network with a source, the vertices of $K$, and a sink. Give each source edge capacity $d\eta$, each orientation of an internal Cayley edge capacity one, and each boundary edge to the sink capacity one. A cut with source-side vertices $F\subseteq K$ has capacity \[ d\eta(|K|-|F|)+|\partial_E F|\ge d\eta|K|. \] The max-flow/min-cut theorem gives a flow saturating all source edges. Subtract opposite internal flows, extend antisymmetrically across the boundary, and set the remaining values to zero. The resulting function has modulus at most one and divergence $d\eta$ on $K$. A diagonal limit along a finite exhaustion of $G$ proves the assertion. \end{proof} Let $\cW_\ell=S^{\times\ell}$, $n_\ell=d^\ell$, and $s_w=s_1\cdots s_\ell$ for $w=(s_1,\ldots,s_\ell)$. Write $\Omega_\ell$ for the set of length-$\ell$ walks. A walk $\omega=(x_0,\ldots,x_\ell)$ has initial vertex $o(\omega)=x_0$, terminal vertex $t(\omega)=x_\ell$, and word $w(\omega)\in\cW_\ell$, with $x_j=x_0s_1\cdots s_j$. Define \begin{equation*} \Phi(\omega)=\sum_{j=1}^{\ell}b(x_{j-1},x_j),\qquad \Omega_\ell^- =\{\Phi\le0\},\qquad \Omega_\ell^+ =\{\Phi>0\}. \end{equation*} Thus $\Omega_\ell=\Omega_\ell^-\sqcup\Omega_\ell^+$. Figure~\ref{fig:walks} shows the two kinds of walks inside a Cayley graph. For $E\subseteq\Omega_\ell$, use the endpoint fibres \[ E_x^o=\{\omega\in E:o(\omega)=x\},\qquad E_y^t=\{\omega\in E:t(\omega)=y\}. \] \begin{figure}[htbp] \centering % BEGIN CAYLEY DIAGRAM \begin{tikzpicture}[x=1.05cm,y=1.05cm,>=Stealth,font=\small, flow/.style={draw=black!38,thin,->}, walknegative/.style={draw=blue!70!black,line width=1.55pt,->}, walkpositive/.style={draw=orange!85!black,line width=1.55pt,->}, vertex/.style={circle,fill=black!70,inner sep=1.3pt}] \coordinate (p) at (0,-1.55); \coordinate (l) at (-3.8,0); \coordinate (o) at (0,0); \coordinate (r) at (3.8,0); \draw[flow,dashed] (0,-2.4)--(p); \node[anchor=north,text=black!60] at (0,-2.4) {towards the fixed end}; \foreach \v in {l,o,r} {\draw[flow] (p)--(\v);} \foreach \prefix/\cx in {l/-3.8,o/0,r/3.8} { \foreach \j/\dx/\dy in {1/-1.2/1.3,2/0/1.7,3/1.2/1.3} { \coordinate (\prefix\j) at (\cx+\dx,\dy); \draw[flow] (\prefix)--(\prefix\j); \foreach \angle in {55,90,125} { \draw[flow,dashed] (\prefix\j)--++(\angle:0.47); } \node[vertex] at (\prefix\j) {}; } } \draw[walknegative] (o1)--(o); \draw[walknegative] (o)--(p); \draw[walkpositive] (p)--(r); \draw[walkpositive] (r)--(r3); \foreach \v in {p,l,o,r} {\node[vertex] at (\v) {};} \node[circle,draw=blue!70!black,fill=white,inner sep=2pt] at (o1) {}; \node[left=4pt,text=blue!70!black] at (o1) {$x$}; \node[circle,draw=black!75,fill=white,inner sep=2pt] at (p) {}; \node[below left=2pt] at (p) {$z$}; \node[circle,draw=orange!85!black,fill=white,inner sep=2pt] at (r3) {}; \node[right=4pt,text=orange!85!black] at (r3) {$y$}; \node[align=left,text=blue!70!black,anchor=east] at (-1.2,-1.75) {$\omega_-:x\longrightarrow z$\\$\Phi(\omega_-)=-2c$}; \node[align=left,text=orange!85!black,anchor=west] at (2.2,-1.3) {$\omega_+:z\longrightarrow y$\\$\Phi(\omega_+)=2c$}; \draw[flow] (-5.3,2.8)--(-4.4,2.8); \node[anchor=west,text=black!65] at (-4.2,2.8) {$b(x,y)=c$ in the arrow direction}; \end{tikzpicture} % END CAYLEY DIAGRAM \caption{A fragment of the $4$-regular Cayley tree of $F_2$. Orient edges away from a fixed end and put $b(x,y)=c$ in that direction, where $0