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Quasi-convex surface subgroups in some one-relator groups with torsion

Andrew Ng

TL;DR

The paper addresses the existence of quasi-convex surface subgroups in one-relator groups with torsion by constructing a surface subgroup in large-torsion quotients $G_n = F_k / \langle\langle w^n \rangle\rangle$ when $w$ is non-primitive. It combines a Wilton-type map-of-pairs from a hyperbolic surface to the free group with Relatively hyperbolic Dehn filling and no-accidents results to realize a surface subgroup inside $G_n$ for all large $n$ in a controlled way, and shows the subgroup is quasi-convex for sufficiently large $n$. The work yields a profinite criterion for primitivity: $w$ is primitive iff, for large $n$, the profinite completion of $G_n$ agrees with that of $K_n = F_{k-1}*\mathbb{Z}/n$ (subject to Remeslennikov’s question). It further characterizes surface-words via the behavior of $G_n$ as $n$ grows and situates the corollaries within the theory of hyperbolic and cubulated groups, using Agol–Groves–Manning and related retract arguments.

Abstract

We find surface subgroups in certain one-relator groups with torsion and use this to deduce a profinite criterion for a word in the free group to be primitive.

Quasi-convex surface subgroups in some one-relator groups with torsion

TL;DR

The paper addresses the existence of quasi-convex surface subgroups in one-relator groups with torsion by constructing a surface subgroup in large-torsion quotients when is non-primitive. It combines a Wilton-type map-of-pairs from a hyperbolic surface to the free group with Relatively hyperbolic Dehn filling and no-accidents results to realize a surface subgroup inside for all large in a controlled way, and shows the subgroup is quasi-convex for sufficiently large . The work yields a profinite criterion for primitivity: is primitive iff, for large , the profinite completion of agrees with that of (subject to Remeslennikov’s question). It further characterizes surface-words via the behavior of as grows and situates the corollaries within the theory of hyperbolic and cubulated groups, using Agol–Groves–Manning and related retract arguments.

Abstract

We find surface subgroups in certain one-relator groups with torsion and use this to deduce a profinite criterion for a word in the free group to be primitive.

Paper Structure

This paper contains 6 sections, 6 theorems.

Table of Contents

  1. Introduction
  2. Background
  3. Relatively hyperbolic Dehn filling
  4. Maps of pairs
  5. Proof of \ref{['surfacesubgroup']}
  6. Proofs of the corollaries

Key Result

Theorem 1

Let $F_k$ be the non-abelian free group with free basis $\{x_1, \dots x_k\}$ and let $w$ be a word in the $x_i$ which isn't primitive. Then there is some positive integer $d$ such that, for sufficiently large $n$ which are multiples of $d$, the one-relator group $G_n:=F_k/\langle\langle w^n\rangle\r

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Theorem 7
  • Definition 8
  • Lemma 9
  • proof
  • ...and 5 more