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Matthew Tointon

Publications and source records attributed to Matthew Tointon.

At least 19 recordsLinked to original sources
arXiv2024

Residually finite groups with uniformly almost flat quotients

We show that if all the finite coset spaces of a polycyclic group have diameter bounded uniformly below by a polynomial in their size then the group is virtually nilpotent. We obtain the same conclusion for a finitely generated residually torsion-free nilpotent group under the weaker assumption that the finite quotient groups have diameter bounded uniformly below by a polynomial in their size. This extends work of Khukhro and Valette.

math.GR↗
arXiv2024

Spread-out percolation on transitive graphs of polynomial growth

Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in $G$. We show that the critical probability $p_c(G_r)$ for Bernoulli bond percolation on $G_r$ satisfies $p_c(G_r) \sim 1/\mathrm{deg}(G_r)$ as $r\to\infty$. This extends work of Penrose and Bollob\'as-Janson-Riordan, who considered the case $G=\mathbb{Z}^d$. Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces a new notion of dimension in groups. It also verifies a special case of a conjecture of Easo and Hutchcroft.

math.PR↗
arXiv2024

Balls in groups: volume, structure and growth

We give sharp bounds in Breuillard, Green and Tao's finitary version of Gromov's theorem on groups with polynomial growth. Precisely, we show that for every non-negative integer d there exists $c=c(d)>0$ such that if $G$ is a group with finite symmetric generating set $S$ containing the identity and $|S^n|\le cn^{d+1}|S|$ for some positive integer $n$ then there exist normal subgroups $H\le\Gamma\le G$ such that $H\subseteq S^n$, such that $\Gamma/H$ is $d$-nilpotent (i.e. has a central series of length $d$ with cyclic factors), and such that $[G:\Gamma]\le g(d)$, where $g(d)$ denotes the maximum order of a finite subgroup of $GL_d(\mathbb{Z})$. The bounds on both the nilpotence and index are sharp; the previous best bounds were $O(d)$ on the nilpotence, and an ineffective function of $d$ on the index. In fact, we obtain this as a small part of a much more detailed fine-scale description of the structure of $G$. These results have a wide range of applications in various aspects of the theory of vertex-transitive graphs: percolation theory, random walks, structure of finite groups, scaling limits of finite vertex-transitive graphs.... We obtain some of these applications in the present paper, and treat others in companion papers. Some are due to or joint with other authors.

math.GR↗
arXiv2023

Small doubling implies small tripling for balls of large radius

We show that if $K\ge1$ is a parameter and $S$ is a finite symmetric subset of a group containing the identity such $|S^{2n}|\le K|S^n|$ for some integer $n\ge2K^2$, then $|S^{3n}|\le\exp(\exp(O(K^2)))|S^n|$. Such a result was previously known only under the stronger assumption that $|S^{2n+1}|\le K|S^n|$. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.

math.CO↗
arXiv2023

New bounds in the discrete analogue of Minkowski's second theorem

We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set $k=\max\{j:\lambda_j\le1\}$, then $K$ contains at most $2^k(1+\frac{\lambda_k}2)^k/\lambda_1\cdots\lambda_k$ lattice points. This provides improved bounds in a conjecture of Betke, Henk and Wills (1993), and verifies that conjecture asymptotically as $\lambda_k\to0$. We also obtain a similar result without the symmetry assumption.

math.MG↗
arXiv2022

Geometric amenability in totally disconnected locally compact groups

We give a short geometric proof of a result of Soardi & Woess and Salvatori that a quasitransitive graph is amenable if and only if its automorphism group is amenable and unimodular. We also strengthen one direction of that result by showing that if a compactly generated totally disconnected locally compact group admits a proper Lipschitz action on a bounded-degree amenable graph then that group is amenable and unimodular. We pass via the notion of geometric amenability of a locally compact group, which has previously been studied by the second author and is defined by analogy with amenability, only using right Folner sets instead of left Folner sets. We also introduce a notion of uniform geometric non-amenability of a locally compact group, and relate this notion in various ways to actions of that group on graphs and to its modular homomorphism.

math.GR↗
arXiv2021

Non-triviality of the phase transition for percolation on finite transitive graphs

We prove that if $(G_n)_{n\geq1}=((V_n,E_n))_{n\geq 1}$ is a sequence of finite, vertex-transitive graphs with bounded degrees and $|V_n|\to\infty$ that is at least $(1+\epsilon)$-dimensional for some $\epsilon>0$ in the sense that \[\mathrm{diam} (G_n)=O\left(|V_n|^{1/(1+\epsilon)}\right) \text{ as $n\to\infty$}\] then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each $0<\alpha <1$ there exists $p_c(\alpha)<1$ such that for each $p\geq p_c(\alpha)$, Bernoulli-$p$ bond percolation on $G_n$ has a cluster of size at least $\alpha |V_n|$ with probability tending to $1$ as $n\to \infty$. In fact, we prove more generally that there exists a universal constant $a$ such that the same conclusion holds whenever \[\mathrm{diam} (G_n)=O\left(\frac{|V_n|}{(\log |V_n|)^a}\right) \text{ as $n\to\infty$.}\] This verifies a conjecture of Benjamini up to the value of the constant $a$, which he suggested should be $1$. We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits and prove that one may indeed take $a=1$ when the graphs $G_n$ are all Cayley graphs of Abelian groups. A key step in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a nontrivial uniqueness phase (i.e., to have $p_u<1$) which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at $1$ in the sense that for every $k,n<\infty$ there exists $\epsilon>0$ such that every infinite graph $G$ of degree at most $k$ whose vertex set has at most $n$ orbits under Aut$(G)$ either has $p_c=1$ or $p_c\leq 1-\epsilon$.

math.PR↗
arXiv2020

Explicit universal minimal constants for polynomial growth of groups

Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and degree of polynomial growth both bounded effectively. The third and fourth authors proved the optimal bound on the degree of polynomial growth of this subgroup, at the expense of making some other parts of the result ineffective. In the present paper we prove the optimal bound on the degree of polynomial growth without making any losses elsewhere. As a consequence, we show that there exist explicit positive numbers $\varepsilon_d$ such that in any group with growth at least a polynomial of degree $d$, the growth is at least $\varepsilon_dn^d$. We indicate some applications in probability; in particular, we show that the gap at $1$ for the critical probability for Bernoulli site percolation on a Cayley graph, recently proven to exist by Panagiotis and Severo, is at least $\exp\bigl\{-\exp\bigl\{17 \exp\{100 \cdot 8^{100}\}\bigr\}\bigr\}$.

math.GR↗
arXiv2020

Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs

We prove sharp bounds on the probability that the simple random walk on a vertex-transitive graph escapes the ball of radius $r$ before returning to its starting point. In particular, this shows that if the ball of radius $r$ has size slightly greater than quadratic in $r$ then this probability is bounded from below. On the other hand, we show that if the ball of radius $r$ has volume slightly less than cubic in $r$ then this probability decays logarithmically for all larger balls. These results represent a finitary refinement of Varopoulos's theorem that a random walk on a vertex-transitive graph is recurrent if and only if the graph has at most quadratic volume growth. They also imply the existence of a gap at $0$ for escape probabilities: there exists a universal constant $c>0$ such that the random walk on an arbitrary vertex-transitive graph is either recurrent or has a probability of at least $c$ of escaping to infinity. We also prove versions of these results for finite graphs, in particular confirming and strengthening a conjecture of Benjamini and Kozma from 2002. Amongst other things, we also generalise our results to give a sharp finitary version of the characterisation of $p$-parabolic vertex-transitive graphs, prove a number of sharp isoperimetric inequalities for vertex-transitive graphs, and prove a locality result for the escape probability of the random walk on a vertex-transitive graph that can be seen as an analogue of Schramm's locality conjecture for the critical percolation probability.

math.PR↗
arXiv2019

A finitary structure theorem for vertex-transitive graphs of polynomial growth

We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph G of large diameter admits a quotient with fibres of small diameter on which the action of Aut(G) is virtually abelian with vertex stabilisers of bounded size. We also show that G has moderate growth in the sense of Diaconis and Saloff-Coste, which is known to imply that the mixing and relaxation times of the lazy random walk on G are quadratic in the diameter. These results extend results of Breuillard and the second author for finite Cayley graphs of large diameter. Finally, given a connected, locally finite vertex-transitive graph G exhibiting polynomial growth at a single, sufficiently large scale, we describe its growth at subsequent scales, extending a result of Tao and an earlier result of our own for Cayley graphs. In forthcoming work we will give further applications.

math.CO↗
arXiv2018

Polylogarithmic bounds in the nilpotent Freiman theorem

We show that if $A$ is a finite $K$-approximate subgroup of an $s$-step nilpotent group then there is a finite normal subgroup $H\subset A^{K^{O_s(1)}}$ modulo which $A^{O_s(\log^{O_s(1)}K)}$ contains a nilprogression of rank at most $O_s(\log^{O_s(1)}K)$ and size at least $\exp(-O_s(\log^{O_s(1)}K))|A|$. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard-Green, Breuillard-Green-Tao, Gill-Helfgott-Pyber-Szab\'o, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.

math.CO↗
arXiv2018

Probabilistic nilpotence in infinite groups

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Folner sequence if G is amenable. In our first main result we show that if G is finitely generated then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent. This generalises both an earlier result of the second author treating the case k=1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below then G is virtually k-step nilpotent, answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients, generalising a result of Gallagher.

math.GR↗
arXiv2017

Scaling limits of Cayley graphs with polynomially growing balls

Benjamini, Finucane and the first author have shown that if (G_n,S_n) is a sequence of Cayley graphs such that |S_n^n|=O(n^D|S_n|), then the sequence (G_n,d_{S_n}/n) is relatively compact for the Gromov-Hausdorff topology and every cluster point is a connected nilpotent Lie group equipped with a left-invariant sub-Finsler metric. In this paper we show that the dimension of such a cluster point is bounded by D, and that, under the stronger bound |S_n^n|=O(n^D), the homogeneous dimension of a cluster point is bounded by D. Our approach is roughly to use a well-known structure theorem for approximate groups due to Breuillard, Green and Tao to replace S_n^n with a coset nilprogression of bounded rank, and then to use results about nilprogressions from a previous paper of ours to study the ultralimits of such coset nilprogressions. As an application we bound the dimension of the scaling limit of a sequence of vertex-transitive graphs of large diameter. We also recover and effectivise parts of an argument of Tao concerning the further growth of single set S satisfying the bound |S^n| < Mn^D|S|.

math.GR↗
arXiv2017

The Mahler conjecture in two dimensions via the probabilistic method

The "Mahler volume" is, intuitively speaking, a measure of how "round" a centrally symmetric convex body is. In one direction this intuition is given weight by a result of Santalo, who in the 1940s showed that the Mahler volume is maximized, in a given dimension, by the unit sphere and its linear images, and only these. A counterpart to this result in the opposite direction is proposed by a conjecture, formulated by Kurt Mahler in the 1930s and still open in dimensions 4 and greater, asserting that the Mahler volume should be minimized by a cuboid. In this article we present a seemingly new proof of the 2-dimensional case of this conjecture via the probabilistic method. The central idea is to show that either deleting a random pair of edges from a centrally symmetric convex polygon, or deleting a random pair of vertices, reduces the Mahler volume with positive probability.

math.MG↗
arXiv2017

Commuting probabilities of infinite groups

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox.

math.GR↗
arXiv2016

Properness of nilprogressions and the persistence of polynomial growth of given degree

We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a Lie-algebra version of the geometry-of-numbers argument at the centre of that result. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1\in S$ and $|S^n|\le Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth of the same degree $D$ at all subsequent scales, in the sense that $|S^r|\ll_{M,D}r^D$ for every $r\ge n$. Our methods also provide an important ingredient in a forthcoming companion paper in which we reprove and sharpen a result about scaling limits of vertex-transitive graphs of polynomial growth due to Benjamini, Finucane and the first author. We also note that our arguments imply that every approximate group has a large subset with a large quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent group of bounded rank and step.

math.GR↗