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Matthew Tointon

## Matthew TointonI am a Senior Lecturer in the School of Mathematics at the University of Bristol. I tend to be interested in problems and results concerning the combinatorial, geometric and probabilistic properties of groups. Much of my work lies at the interface between additive combinatorics and geometric group theory. |

## Book: Introduction to Approximate GroupsMy bookIntroduction to Approximate Groups is now available to read or buy online.
London Mathematical Society Student Texts Reviewed by Michael Björklund in |

In reverse chronological order:

- Spread-out percolation on transitive graphs of polynomial growth (with Panagiotis Spanos), 35pp, submitted. arXiv
- Balls in groups: volume, structure and growth (with Romain Tessera), 94pp, submitted. arXiv
- Small doubling implies small tripling at large scales (with Romain Tessera), 5pp, submitted. arXiv
- New bounds in the discrete analogue of Minkowski's second theorem, 5pp, to appear in
*Discrete Anal.*. arXiv - Geometric amenability in totally disconnected locally compact groups (with Romain Tessera),
*Math. Z.*.**304**(2023), article no 4, 19pp. arXiv | Journal - Explicit universal minimal constants for polynomial growth of groups (with Russell Lyons, Avinoam Mann and Romain Tessera),
*J. Group Theory***26**(2023), 29-53. arXiv | Journal - Non-triviality of the phase transition for percolation on finite transitive graphs (with Tom Hutchcroft), 62pp, to appear in
*J. Eur. Math. Soc.*. arXiv - Sharp relations between volume growth, isoperimetry and resistance in vertex-transitive graphs (with Romain Tessera), 35pp. arXiv
- A finitary structure theorem for vertex-transitive graphs of polynomial growth (with Romain Tessera),
*Combinatorica***41**(2021), 263-298. arXiv | Journal - Polylogarithmic bounds in the nilpotent Freiman theorem,
*Math. Proc. Cambridge Philos. Soc.***170**(2021), 111-127. arXiv | Journal - Probabilistic nilpotence in infinite groups (with Armando Martino, Motiejus Valiunas and Enric Ventura),
*Israel J. Math.***244**(2021), 539-588. arXiv | Journal - Scaling limits of Cayley graphs with polynomially growing balls (with Romain Tessera), 37pp. arXiv
- The Mahler conjecture in two dimensions via the probabilistic method,
*Amer. Math. Monthly***125**(2018), 820-828. arXiv | Journal - Commuting probabilities of infinite groups,
*J. London Math. Soc.***101**(2020), 1280-1297. arXiv | Journal - The asymptotic dimension of box spaces of virtually nilpotent groups (with Thiebout Delabie),
*Discrete Math.***341**(2018), 1036-1040. arXiv | Journal - Properness of nilprogressions and the persistence of polynomial growth of given degree (with Romain Tessera),
*Discrete Anal.*2018:17, 38 pp. arXiv | Journal - Horofunctions on graphs of linear growth (with Ariel Yadin),
*C. R. Math. Acad. Sci. Paris***354**(2016), 1151-1154. arXiv | Journal - Approximate subgroups of residually nilpotent groups,
*Math. Ann.***374**(2019), 499-515. arXiv | Journal - Nilprogressions and groups with moderate growth (with Emmanuel Breuillard),
*Adv. Math.***289**(2016), 1008-1055. arXiv | Journal - Polynomials and harmonic functions on discrete groups (with Tom Meyerovitch, Idan Perl and Ariel Yadin),
*Trans. Amer. Math. Soc.***369**(2017), 2205-2229. arXiv | Journal - Characterisations of algebraic properties of groups in terms of harmonic functions,
*Groups Geom. Dyn.***10**(2016), 1007-1049. arXiv | Journal - Recurrence and non-uniformity of bracket polynomials,
*Online J. Anal. Comb.***9**(2014), 36pp. arXiv | Journal - Freiman's theorem in an arbitrary nilpotent group,
*Proc. London Math. Soc.***109**(2014), 318-352. arXiv | Journal

In reverse chronological order:

- A brief introduction to approximate groups,
*Eur. Math. Soc. Newsl.***115**(2020), 12-16. arXiv | Journal - Raconte-moi... les groupes approximatifs (in French),
*Gaz. Math.***160**(2019), 53-59. arXiv | Journal

- Percolation on finite transitive graphs, Webinar in Additive Combinatorics, 2nd March 2022.
- Probability on finite transitive graphs, Joint Israeli Probability Seminar, 13th December 2021.
- Non-triviality of the phase transition for percolation on finite transitive graphs (joint talk with Tom Hutchcroft), Percolation Today, 11th May 2021.

The following video is a non-technical description of parts of my work.

An alternative approach to Freiman's theorem in *p*-groups.

An essay on the Tits alternative, which I wrote in 2009 as part of my Part III studies in Cambridge.

Graduate student and postdoc seminar 2015/16, Université Paris-Sud