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Isometry groups of infinite-genus hyperbolic surfaces

Tarik Aougab, Priyam Patel, Nicholas G. Vlamis

Abstract

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

Isometry groups of infinite-genus hyperbolic surfaces

Abstract

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

Paper Structure

This paper contains 19 sections, 58 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological ends
  4. The classification of surfaces
  5. Countable compact Hausdorff spaces
  6. Hyperbolic geometry
  7. Finite groups
  8. The Loch Ness monster surface
  9. Denumerable groups
  10. Non-displaceable subsurfaces
  11. Radial symmetry
  12. Doubly pointed spaces
  13. Radial symmetry and self-similarity
  14. Self-similarity implies radial symmetry
  15. Radial symmetry implies self-similarity
  16. ...and 4 more sections

Key Result

Corollary 1.2

If $S$ is an orientable infinite-genus 2-manifold with no planar ends, then $\mathop{\mathrm{Homeo}}\nolimits(S), \mathop{\mathrm{Diffeo}}\nolimits(S),$ and $\mathop{\mathrm{MCG}}\nolimits(S)$ each contain an isomorphic copy of every finite group.

Figures (5)

  • Figure 1: Examples for the three cases of Theorems \ref{['mainUncountable']} and \ref{['main']}.
  • Figure 2: For $S$ with $|\mathcal{E}|=3$, we have the curves $c_0, c_1$, the neighborhoods $\nu_0, \nu_1$, and the surfaces $S_1, S_2,$ and $S_3$.
  • Figure 3: The map $\varphi_1$, where we imagine the point at $\infty$ of $Z_2$ is sent to the unique end of $S_1$.
  • Figure 4: On the left hand side is $Z^G_2$, where $|G| = 4$, and on the right hand side is $R_\mathcal{E}$.
  • Figure 5: An example of the base case for recursion, where $V_1\smallsetminus V_2 = B_ 1 \sqcup B_2 \sqcup B_3$.

Theorems & Definitions (95)

  • Remark 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3: Classification of orientable 2-manifolds
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 85 more