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Soluble quotients of triangle groups

Marston D. E. Conder, Darius W. Young

Abstract

This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for `small' genus), by showing that every non-perfect hyperbolic ordinary triangle group $Δ^+(p,q,r) = \langle\, x,y \ | \ x^p = y^q = (xy)^r = 1 \,\rangle$ has a smooth finite soluble quotient of derived length $c$ for some $c \le 3$, and infinitely many such quotients of derived length $d$ for every $d > c$.

Soluble quotients of triangle groups

Abstract

This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for `small' genus), by showing that every non-perfect hyperbolic ordinary triangle group has a smooth finite soluble quotient of derived length for some , and infinitely many such quotients of derived length for every .

Paper Structure

This paper contains 5 sections, 5 theorems, 18 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Further background
  3. The derived subgroup of a non-perfect triangle group
  4. Proof of our main theorem
  5. Final remarks

Key Result

Theorem 1.1

If $\Delta^+$ is any non-perfect hyperbolic ordinary triangle group, then (a) $\Delta^+$ has a smooth finite soluble quotient with derived length at most $3$, (b) if $c$ is the minimum derived length of a smooth finite soluble quotient of $\Delta^+$, then $\Delta^+$ has infinitely many smooth finite

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1
  • Example 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 5.1
  • proof