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Matthew Tointon

I am the Stokes Research Fellow at Pembroke College, Cambridge, affiliated to the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. From July 2020 I will be 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. A brief introduction to approximate groups, Eur. Math. Soc. Newsl. 115 (2020), 12-16.     arXiv    |    Journal
  2. Sharp relations between volume growth, isoperimetry and resistance in vertex-transitive graphs (with Romain Tessera), 35pp.     arXiv
  3. A finitary structure theorem for vertex-transitive graphs of polynomial growth (with Romain Tessera), to appear in Combinatorica, 25pp.     arXiv
  4. Raconte-moi... les groupes approximatifs (in French), Gaz. Math. 160 (2019), 53-59.     arXiv    |    Journal
  5. Polylogarithmic bounds in the nilpotent Freiman theorem, to appear in Math. Proc. Cambridge Philos. Soc., 15pp     arXiv    |    Journal
  6. Probabilistic nilpotence in infinite groups (with Armando Martino, Motiejus Valiunas and Enric Ventura), to appear in Israel J. Math., 31pp.     arXiv
  7. Scaling limits of Cayley graphs with polynomially growing balls (with Romain Tessera), 37pp.     arXiv
  8. The Mahler conjecture in two dimensions via the probabilistic method, Amer. Math. Monthly 125 (2018), 820-828.     arXiv    |    Journal
  9. Commuting probabilities of infinite groups, J. London Math. Soc. 101 (2020), 1280-1297.     arXiv    |    Journal
  10. The asymptotic dimension of box spaces of virtually nilpotent groups (with Thiebout Delabie), Discrete Math. 341 (2018), 1036-1040.     arXiv    |    Journal
  11. Properness of nilprogressions and the persistence of polynomial growth of given degree (with Romain Tessera), Discrete Anal. 2018:17, 38 pp.     arXiv    |    Journal
  12. Horofunctions on graphs of linear growth (with Ariel Yadin), C. R. Math. Acad. Sci. Paris 354 (2016), 1151-1154.     arXiv    |    Journal
  13. Approximate subgroups of residually nilpotent groups, Math. Ann. 374 (2019), 499-515.     arXiv    |    Journal
  14. Nilprogressions and groups with moderate growth (with Emmanuel Breuillard), Adv. Math. 289 (2016), 1008-1055.     arXiv    |    Journal
  15. 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
  16. Characterisations of algebraic properties of groups in terms of harmonic functions, Groups Geom. Dyn. 10 (2016), 1007-1049.     arXiv    |    Journal
  17. Recurrence and non-uniformity of bracket polynomials, Online J. Anal. Comb. 9 (2014), 36pp.     arXiv    |    Journal
  18. Freiman's theorem in an arbitrary nilpotent group, Proc. London Math. Soc. 109 (2014), 318-352.     arXiv    |    Journal


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


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