Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chasing finite shadows of infinite groups through geometry

Martin R. Bridson

TL;DR

The notes survey how infinite groups can be probed through their finite quotients, emphasizing profinite completions, residual finiteness, and three notions of rigidity (absolute, relative, Grothendieck). They weave together algorithmic barriers, geometric group theory, and low-dimensional topology to show how hyperbolic geometry, CAT$(0)$ cube complexes, and arithmetic structures drive strong rigidity phenomena, including absolute rigidity for select Kleinian and Fuchsian groups and Grothendieck rigidity phenomena. A central theme is that finiteness properties and geometric decompositions (e.g. JSJ, Seifert fibrations) often leave imprints in profinite data, enabling sharp distinctions among groups and, in favorable cases, complete determination by finite quotients. The work also outlines powerful constructions (Rips, fiber products, 1–2–3 theorem) that generate Grothendieck pairs, highlighting both the potential and limits of profinite methods, and closes with key open questions about universal rigidity, free groups, and higher-rank lattices.

Abstract

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects, i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its finite images? Which properties of the group can one recognise, and when does the set of finite images determine the group completely? How hard is it to decide what the finite images of a given infinite group are? These notes follow my plenary lecture at the ECM in Sevilla, July 2024. The goal of the lecture was to sketch some of the rich history of the preceding problems and to present results that illustrate how the field surrounding these questions has been transformed in recent years by input from low-dimensional topology and the study of non-positively curved spaces.

Chasing finite shadows of infinite groups through geometry

TL;DR

The notes survey how infinite groups can be probed through their finite quotients, emphasizing profinite completions, residual finiteness, and three notions of rigidity (absolute, relative, Grothendieck). They weave together algorithmic barriers, geometric group theory, and low-dimensional topology to show how hyperbolic geometry, CAT cube complexes, and arithmetic structures drive strong rigidity phenomena, including absolute rigidity for select Kleinian and Fuchsian groups and Grothendieck rigidity phenomena. A central theme is that finiteness properties and geometric decompositions (e.g. JSJ, Seifert fibrations) often leave imprints in profinite data, enabling sharp distinctions among groups and, in favorable cases, complete determination by finite quotients. The work also outlines powerful constructions (Rips, fiber products, 1–2–3 theorem) that generate Grothendieck pairs, highlighting both the potential and limits of profinite methods, and closes with key open questions about universal rigidity, free groups, and higher-rank lattices.

Abstract

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects, i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its finite images? Which properties of the group can one recognise, and when does the set of finite images determine the group completely? How hard is it to decide what the finite images of a given infinite group are? These notes follow my plenary lecture at the ECM in Sevilla, July 2024. The goal of the lecture was to sketch some of the rich history of the preceding problems and to present results that illustrate how the field surrounding these questions has been transformed in recent years by input from low-dimensional topology and the study of non-positively curved spaces.

Paper Structure

This paper contains 39 sections, 25 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Presenting Groups and Understanding Them
  3. Groups everywhere
  4. How should we capture groups?
  5. Presentations of Groups
  6. Groups that do not have finite presentations
  7. Problem: hard to extract information from a presentation
  8. We need action!
  9. A group springing into action: topological models
  10. Decision Problems (Max Dehn 1911)
  11. Unsolvable Problems
  12. 3-manifold groups are well behaved
  13. Returning to the groups $G_2, \ G_3, \ G_4$
  14. Really useful actions: Trees, hyperbolicity, and cube complexes
  15. Hard to see if finite images exist
  16. ...and 24 more sections

Key Result

Theorem 1

There does not exist an algorithm that, given an arbitrary finite presentation, can determine whether or not the group presented has a non-trivial finite quotient.

Figures (4)

  • Figure 1: $\Delta(3, 3, 4) = \langle A, B, C \mid A^3=B^3=C^4=1=ABC\rangle$
  • Figure 2: $\langle T, F \mid T^3=1=F^2, \ FTF=T^2\rangle.$
  • Figure 3: The 2-complex creating an action of $\mathbb Z^2=\langle a, b \mid ab=ba \rangle$
  • Figure 4: A 2-dimensional CAT$(0)$ cube complex

Theorems & Definitions (40)

  • Theorem 1: BW
  • Theorem 2: BG
  • Theorem 3: BMRS2
  • Theorem 4: BRS
  • Example 2.1
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5: Noviko-Boone
  • Theorem 2.6
  • Theorem 3.1: Bridson, Wilton BW
  • ...and 30 more