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Characterizing closed curves on Riemann surfaces via homology groups of coverings

Marco Boggi, Pavel Zalesskii

Abstract

Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally determine in the homology groups of the compactifications of unramified $p$-coverings of $S$, for $p\geq 2$ a fixed prime. In particular, this gives a characterization of simple closed curves on $S$ in terms of homology groups of $p$-coverings. We then define a $p$-adic Reidemeister pairing on the fundamental group of $S$ and show that the free homotopy classes of two loops have trivial geometric intersection if and only if they are orthogonal with respect to this pairing. As an application, we give a geometric argument to prove that oriented surface groups are conjugacy $p$-separable (a combinatorial proof of this fact was recentely given by Paris).

Characterizing closed curves on Riemann surfaces via homology groups of coverings

Abstract

Let be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on is detected by some symplectic submodules they naturally determine in the homology groups of the compactifications of unramified -coverings of , for a fixed prime. In particular, this gives a characterization of simple closed curves on in terms of homology groups of -coverings. We then define a -adic Reidemeister pairing on the fundamental group of and show that the free homotopy classes of two loops have trivial geometric intersection if and only if they are orthogonal with respect to this pairing. As an application, we give a geometric argument to prove that oriented surface groups are conjugacy -separable (a combinatorial proof of this fact was recentely given by Paris).

Paper Structure

This paper contains 6 sections, 15 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Characterization of simple closed curves
  3. The pro-${\mathscr C}$ hyperbolic solenoid
  4. An algebraic characterization of simple closed curves via a pro-${\mathscr C}$ Reidemeister pairing
  5. Distinguishing closed curves
  6. Appendix on pro-$p$ surface groups

Key Result

Theorem \oldthetheorem

Let ${\mathscr C}$ be a class of finite groups. Then, a pair of (not necessarily distinct) closed curves $\gamma$ and $\gamma'$ on a Riemann surface $S_{g,n}$ have trivial geometric intersection if and only if, for a cofinal system of ${\mathscr C}$-open subgroups $\{K\}$ of $\Pi_{g,n}$, there holds

Theorems & Definitions (32)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • ...and 22 more