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Matthew Tointon

I 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 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:

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

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


Emmanuel Breuillard, Thiebout Delabie, Tom Hutchcroft, Russell Lyons, Avinoam Mann, Armando Martino, Tom Meyerovitch, Idan Perl, Panagiotis Spanos, 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