arXiv subjects

Matthew Tointon

Publications and source records attributed to Matthew Tointon.

26 records · Page 2Linked to original sources

arXiv2018

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↗

arXiv2016

Horofunctions on graphs of linear growth

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 cyclic.

math.MG↗

arXiv2016

Polynomials and harmonic functions on discrete groups

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 finite index is always a polynomial in the Mal'cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos's result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at most k is finite dimensional. We also prove that for a finitely generated group the Laplacian maps the polynomials of degree k surjectively onto the polynomials of degree k - 2. We then present some corollaries. In particular, we calculate precisely the dimension of the space of harmonic functions of polynomial growth of degree at most k on a virtually nilpotent group, extending an old result of Heilbronn for the abelian case, and refining a more recent result of Hua and Jost.

math.GR↗

arXiv2016

Characterisations of algebraic properties of groups in terms of harmonic functions

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 with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite graph is surjective; and we show that the positive harmonic functions on a non-virtually nilpotent linear group span an infinite-dimensional space.

math.GR↗

arXiv2015

Nilprogressions and groups with moderate growth

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 proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter. We also give bounds on the Cheeger constant, first eigenvalue of the Laplacian, and mixing time. This can be seen as a finite-group version of Gromov's theorem on groups with polynomial growth. It also improves on a result of Lackenby regarding property (tau) in towers of coverings. Another consequence is a universal upper bound on the diameter of all finite simple groups, independent of the CFSG.

math.GR↗

arXiv2014

Recurrence and non-uniformity of bracket polynomials

A bracket polynomial on the integers is a function formed using the operations of addition, multiplication and taking fractional parts. For a fairly large class of bracket polynomials we show that if p is a bracket polynomial of degree k-1 on [N] then the function f defined by f(n) = e(p(n)) has Gowers U^k[N]-norm bounded away from zero, uniformly in N. We establish this result by first reducing it to a certain recurrence property of sets of bracket polynomials. Specifically, for a fairly large class of bracket polynomials we show that if p_1, ..., p_r are bracket polynomials then their values, modulo 1, are all close to zero on at least some constant proportion of the points 1, ..., N. The proofs rely on two deep results from the literature. The first is work of V. Bergelson and A. Leibman showing that an arbitrary bracket polynomial can be expressed in terms of a polynomial sequence on a nilmanifold. The second is a theorem of B. Green and T. Tao describing the quantitative distribution properties of such polynomial sequences. We give elementary alternative proofs of the first result, without reference to nilmanifolds, in certain "low-complexity" special cases.

math.NT↗

arXiv2014

Freiman's theorem in an arbitrary nilpotent group

We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate subgroup A of an s-step nilpotent group G is contained in a coset nilprogression of rank at most f(K) and cardinality at most exp(g(K))|A|, with f and g polynomials depending only on the step s of G. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem.

math.CO↗

arXiv2013

On harmonic functions and the linear-growth case of Gromov's theorem

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 integers. A key tool is Wilkie and van den Dries's quantitative version of the linear-growth case of Gromov's theorem on groups of polynomial growth.

math.GR↗