Teaching

Teaching materials may be found here.

Research interests

My work mostly centres around interactions of the worlds of geometric group theory and geometric topology with profinite groups. This involves both the study of residual properties and profinite completions of groups and the extension of geometric concepts to the category of profinite or pro-p groups.

My books

Profinite Groups and Residual Finiteness, European Mathematical Society, August 2024.

This book describes the theory of profinite groups, from the basics of the theory to topics which are active areas of current research. It is the first textbook on profinite groups to make their use in studying residually finite groups via their profinite completions a central focus.

The first part of the book gives the subject a firm theoretical underpinning from category theory and introduces profinite groups as objects worthy of study in their own right. The reader is not expected to have a background in category theory. The connection of a residually finite group to its profinite completion is explored in detail, with emphasis on various separability properties and profinite rigidity.

The study of group cohomology is a key tool in the exploration of profinite groups. The central portion of this book gives a standalone first course in group cohomology before showing the modification of this theory for use with profinite groups. There is special emphasis on the unique features of profinite group cohomology such as Pontrjagin duality and Sylow theory.

Later chapters of the book collect together for the first time important results concerning the relation of the cohomology of a group to that of its profinite completion, and introduce the concept of an action of a profinite group on a profinite tree. This material aims to be a useful reference for researchers as well as a learning resource.

Publications

  1. On the structure of vertex cuts separating the ends of a graph. Pacific J. Math. (Volume 278 Issue 2, Dec 2015).
  2. Profinite properties of RAAGs and special groups. Joint with Robert Kropholler. Bull. Lond. Math. Soc. (Volume 48 Issue 6, Sept 2016).
  3. Profinite rigidity for Seifert fibre spaces. Geom. Dedicata (Volume 188 Issue 1, June 2017).
  4. Virtual pro-p properties of 3-manifold groups. J. Group Theory (Volume 20 Issue 5, Sept 2017).
  5. Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds. J. Algebra (Volume 502, May 2018).
  6. Profinite completions, cohomology and JSJ decompositions of compact 3-manifolds. New Zealand J. Math. (Volume 48, 2018).
  7. Relative cohomology theory for profinite groups. J. Pure and Appl. Algebra (Volume 223 Issue 4, April 2019).
  8. On accessibility for pro-p groups. J. Algebra (Volume 525, May 2019).
  9. Profinite rigidity of graph manifolds, II: knots and mapping classes. Israel J. Math. (Volume 233 Issue 1, August 2019).
  10. A criterion for residual p-finiteness of arbitrary graphs of finite p-groups. J. Group Theory (Volume 22 Issue 4, September 2019).
  11. Classification of pro-p PD2 pairs and the pro-p curve complex. Groups Geom. Dyn. (Volume 4, Issue 3, October 2020).
  12. Distinguishing crystallographic groups by their finite quotients. Joint with Pawel Piwek and David Popovic. J. Algebra (Volume 565, January 2021).
  13. Virtually abelian quotients of random groups. Preprint Feb 2019.
  14. A sufficient condition for accessibility of pro-p groups. Preprint July 2020.
  15. L2 Betti numbers and coherence of random groups. Joint with Dawid Kielak and Robert Kropholler. J. Lond. Math. Soc. (Volume 106, Issue 1, July 2022).
  16. Relative extensions and cohomology of profinite groups. J. Pure and Appl. Algebra (Volume 229 Issue 1, January 2025).
  17. Pontryagin duaity and sheaves of profinite modules. Preprint September 2024.

My doctoral thesis

Profinite Properties of 3-Manifold Groups, Oxford 2018