Matthew Tointon I am an Associate Professor 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 Groups My book Introduction to Approximate Groups is now available to read or buy online. London Mathematical Society Student Texts 94, Cambridge University Press, 2020. Reviewed by Michael Björklund in Bull. Amer. Math. Soc..

Papers and preprints

In reverse chronological order:

24. Residually finite groups with uniformly almost flat quotients (with David Guo), 35pp.     arXiv
23. Spread-out percolation on transitive graphs of polynomial growth (with Panagiotis Spanos), 35pp, submitted.     arXiv
22. Balls in groups: volume, structure and growth (with Romain Tessera), 94pp, submitted.     arXiv
21. Small doubling implies small tripling at large scales (with Romain Tessera), 8pp, submitted.     arXiv
20. New bounds in the discrete analogue of Minkowski's second theorem, 5pp, to appear in Discrete Anal..     arXiv    |    Journal
19. Geometric amenability in totally disconnected locally compact groups (with Romain Tessera), Math. Z.. 304 (2023), article no 4, 19pp.     arXiv    |    Journal
18. 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
17. Non-triviality of the phase transition for percolation on finite transitive graphs (with Tom Hutchcroft), 62pp, to appear in J. Eur. Math. Soc..     arXiv    |    Journal
16. Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs (with Romain Tessera), 36pp, submitted.     arXiv
15. A finitary structure theorem for vertex-transitive graphs of polynomial growth (with Romain Tessera), Combinatorica 41 (2021), 263-298.     arXiv    |    Journal
14. Polylogarithmic bounds in the nilpotent Freiman theorem, Math. Proc. Cambridge Philos. Soc. 170 (2021), 111-127.     arXiv    |    Journal
13. Probabilistic nilpotence in infinite groups (with Armando Martino, Motiejus Valiunas and Enric Ventura), Israel J. Math. 244 (2021), 539-588.     arXiv    |    Journal
12. Scaling limits of Cayley graphs with polynomially growing balls (with Romain Tessera), 37pp.     arXiv
11. The Mahler conjecture in two dimensions via the probabilistic method, Amer. Math. Monthly 125 (2018), 820-828.     arXiv    |    Journal
10. Commuting probabilities of infinite groups, J. London Math. Soc. 101 (2020), 1280-1297.     arXiv    |    Journal
9. The asymptotic dimension of box spaces of virtually nilpotent groups (with Thiebout Delabie), Discrete Math. 341 (2018), 1036-1040.     arXiv    |    Journal
8. Properness of nilprogressions and the persistence of polynomial growth of given degree (with Romain Tessera), Discrete Anal. 2018:17, 38 pp.     arXiv    |    Journal
7. Horofunctions on graphs of linear growth (with Ariel Yadin), C. R. Math. Acad. Sci. Paris 354 (2016), 1151-1154.     arXiv    |    Journal
6. Approximate subgroups of residually nilpotent groups, Math. Ann. 374 (2019), 499-515.     arXiv    |    Journal
5. Nilprogressions and groups with moderate growth (with Emmanuel Breuillard), Adv. Math. 289 (2016), 1008-1055.     arXiv    |    Journal
4. 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
3. Characterisations of algebraic properties of groups in terms of harmonic functions, Groups Geom. Dyn. 10 (2016), 1007-1049.     arXiv    |    Journal
2. Recurrence and non-uniformity of bracket polynomials, Online J. Anal. Comb. 9 (2014), 36pp.     arXiv    |    Journal
1. Freiman's theorem in an arbitrary nilpotent group, Proc. London Math. Soc. 109 (2014), 318-352.     arXiv    |    Journal

Surveys and expository works

In reverse chronological order:

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

Coauthors

Seminar videos

• 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.

Reviews

Video

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

Unpublished material

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.