13 citations · 29 across the 17 of their papers we have counts for
25 papers
Residually finite groups with uniformly almost flat quotients
David Guo, Matthew Tointon
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 obta…
Spread-out percolation on transitive graphs of polynomial growth
Panagiotis Spanos, Matthew Tointon
Let be a vertex-transitive graph of superlinear polynomial growth. Given , let be the graph on the same vertex set as , with two vertices joined by an edge if and…
Balls in groups: volume, structure and growth
Romain Tessera, Matthew Tointon
We develop a fine-scale structural theory for balls of polynomial volume in finitely generated groups. Our main result shows that a polynomial-volume hypothesis on a single ball im…
Small doubling implies small tripling for balls of large radius
Romain Tessera, Matthew Tointon
We show that if is a parameter and is a finite symmetric subset of a group containing the identity such for some integer , then $|S^{3n}|…
New bounds in the discrete analogue of Minkowski's second theorem
Matthew Tointon
We adapt an argument of Tao and Vu to show that if are the successive minima of an origin-symmetric convex body with respect to some lattice $Λ<\mathbb{R}^…
Geometric amenability in totally disconnected locally compact groups
Romain Tessera, Matthew Tointon
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 unimodula…
Non-triviality of the phase transition for percolation on finite transitive graphs
Tom Hutchcroft, Matthew Tointon
We prove that if is a sequence of finite, vertex-transitive graphs with bounded degrees and that is at least -dimensi…
Explicit universal minimal constants for polynomial growth of groups
Russell Lyons, Avinoam Mann, Romain Tessera +1
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 ind…
Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs
Romain Tessera, Matthew Tointon
We prove sharp bounds on the probability that the simple random walk on a vertex-transitive graph escapes the ball of radius before returning to its starting point. In particul…
A finitary structure theorem for vertex-transitive graphs of polynomial growth
Romain Tessera, Matthew Tointon
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…
A brief introduction to approximate groups
Matthew Tointon
We give a brief introduction to the notion of an 'approximate group' and some of its numerous applications.
Polylogarithmic bounds in the nilpotent Freiman theorem
Matthew Tointon
We show that if is a finite -approximate subgroup of an -step nilpotent group then there is a finite normal subgroup modulo which $A^{O_s(\log^{…
Probabilistic nilpotence in infinite groups
Armando Martino, Matthew Tointon, Motiejus Valiunas +1
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…
Scaling limits of Cayley graphs with polynomially growing balls
Romain Tessera, Matthew Tointon
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…
The Mahler conjecture in two dimensions via the probabilistic method
Matthew Tointon
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,…
Commuting probabilities of infinite groups
Matthew Tointon
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 co…
The asymptotic dimension of box spaces of virtually nilpotent groups
Thiebout Delabie, Matthew Tointon
We show that every box space of a virtually nilpotent group has asymptotic dimension equal to the Hirsch length of that group.
Properness of nilprogressions and the persistence of polynomial growth of given degree
Romain Tessera, Matthew Tointon
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…
Horofunctions on graphs of linear growth
Matthew Tointon, Ariel Yadin
We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually…
Approximate subgroups of residually nilpotent groups
Matthew Tointon
We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is…
Nilprogressions and groups with moderate growth
Emmanuel Breuillard, Matthew Tointon
We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups…
Polynomials and harmonic functions on discrete groups
Tom Meyerovitch, Idan Perl, Matthew Tointon +1
Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finit…
Characterisations of algebraic properties of groups in terms of harmonic functions
Matthew Tointon
We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group…
On harmonic functions and the linear-growth case of Gromov's theorem
Matthew Tointon
We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integer…