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Matthew Tointon

I am a 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 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.

Papers and preprints

In reverse chronological order:

  1. Non-triviality of the phase transition for percolation on finite transitive graphs (with Tom Hutchcroft), 60pp, submitted.     arXiv
  2. A brief introduction to approximate groups, Eur. Math. Soc. Newsl. 115 (2020), 12-16.     arXiv    |    Journal
  3. Sharp relations between volume growth, isoperimetry and resistance in vertex-transitive graphs (with Romain Tessera), 35pp.     arXiv
  4. A finitary structure theorem for vertex-transitive graphs of polynomial growth (with Romain Tessera), Combinatorica 41 (2021), 263-298.     arXiv    |    Journal
  5. Raconte-moi... les groupes approximatifs (in French), Gaz. Math. 160 (2019), 53-59.     arXiv    |    Journal
  6. Polylogarithmic bounds in the nilpotent Freiman theorem, Math. Proc. Cambridge Philos. Soc. 170 (2021), 111-127.     arXiv    |    Journal
  7. Probabilistic nilpotence in infinite groups (with Armando Martino, Motiejus Valiunas and Enric Ventura), to appear in Israel J. Math., 31pp.     arXiv
  8. Scaling limits of Cayley graphs with polynomially growing balls (with Romain Tessera), 37pp.     arXiv
  9. 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
  11. The asymptotic dimension of box spaces of virtually nilpotent groups (with Thiebout Delabie), Discrete Math. 341 (2018), 1036-1040.     arXiv    |    Journal
  12. Properness of nilprogressions and the persistence of polynomial growth of given degree (with Romain Tessera), Discrete Anal. 2018:17, 38 pp.     arXiv    |    Journal
  13. Horofunctions on graphs of linear growth (with Ariel Yadin), C. R. Math. Acad. Sci. Paris 354 (2016), 1151-1154.     arXiv    |    Journal
  14. Approximate subgroups of residually nilpotent groups, Math. Ann. 374 (2019), 499-515.     arXiv    |    Journal
  15. Nilprogressions and groups with moderate growth (with Emmanuel Breuillard), Adv. Math. 289 (2016), 1008-1055.     arXiv    |    Journal
  16. 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
  17. Characterisations of algebraic properties of groups in terms of harmonic functions, Groups Geom. Dyn. 10 (2016), 1007-1049.     arXiv    |    Journal
  18. Recurrence and non-uniformity of bracket polynomials, Online J. Anal. Comb. 9 (2014), 36pp.     arXiv    |    Journal
  19. Freiman's theorem in an arbitrary nilpotent group, Proc. London Math. Soc. 109 (2014), 318-352.     arXiv    |    Journal


Emmanuel Breuillard, Thiebout Delabie, Tom Hutchcroft, Armando Martino, Tom Meyerovitch, Idan Perl, Romain Tessera, Motiejus Valiunas, Enric Ventura, Ariel Yadin.

Seminar videos


MathSciNet (subscribers only)


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.

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