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One-endedness of outer automorphism groups of free products of finite and cyclic groups

Rylee Alanza Lyman

TL;DR

This work analyzes outer automorphism groups of free products $F= A_1*\cdots*A_n*F_k$, establishing semistability at infinity for $\mathrm{Out}(F)$ under end-conditions and proving that $H^2(\mathrm{Out}(F),\mathbb{Z}F)$ is free abelian. The approach adapts Vogtmann’s methods on the spine $L(F)$ of reduced Outer Space, introducing a height function via marked graphs of groups and exploiting Whitehead moves, ideal edges, and blowing up operations to push loops to infinity. The paper also classifies the number of ends of $\mathrm{Out}(F)$ in terms of $(n,k)$, showing finite ends only in a narrow regime, infinite ends in a few cases, and otherwise one-ended; this yields partial results towards understanding virtual duality properties of these groups. The results illuminate the structure of the spine $L(F)$, provide a framework for beyond-three-factor free products, and have implications for the cohomology and duality theory of outer automorphism groups of free products.

Abstract

The main result of this paper is that the outer automorphism group of a free product of finite groups and cyclic groups is semistable at infinity (provided it is one ended) or semistable at each end. In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. We also prove that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism groups of the free product of four finite groups or the free product of a single finite group with a free group of rank two are virtual duality groups of dimension two, in contrast with the above example. Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krstić and Vogtmann.

One-endedness of outer automorphism groups of free products of finite and cyclic groups

TL;DR

This work analyzes outer automorphism groups of free products , establishing semistability at infinity for under end-conditions and proving that is free abelian. The approach adapts Vogtmann’s methods on the spine of reduced Outer Space, introducing a height function via marked graphs of groups and exploiting Whitehead moves, ideal edges, and blowing up operations to push loops to infinity. The paper also classifies the number of ends of in terms of , showing finite ends only in a narrow regime, infinite ends in a few cases, and otherwise one-ended; this yields partial results towards understanding virtual duality properties of these groups. The results illuminate the structure of the spine , provide a framework for beyond-three-factor free products, and have implications for the cohomology and duality theory of outer automorphism groups of free products.

Abstract

The main result of this paper is that the outer automorphism group of a free product of finite groups and cyclic groups is semistable at infinity (provided it is one ended) or semistable at each end. In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. We also prove that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism groups of the free product of four finite groups or the free product of a single finite group with a free group of rank two are virtual duality groups of dimension two, in contrast with the above example. Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krstić and Vogtmann.
Paper Structure (24 sections, 22 theorems, 53 equations, 2 figures)

This paper contains 24 sections, 22 theorems, 53 equations, 2 figures.

Table of Contents

  1. Duality groups.
  2. Ends of groups.
  3. End invariants of groups.
  4. Semistability.
  5. The case of $\mathop{\mathrm{Out}}\nolimits(F)$.
  6. Acknowledgments.
  7. The Complex L
  8. The complex $L$.
  9. The norm $\|\cdot\|$.
  10. Directions and turns.
  11. The star graph.
  12. Ideal edges.
  13. Blowing up ideal edges.
  14. The absolute value of an ideal edge.
  15. Connectivity at Infinity
  16. ...and 9 more sections

Key Result

Theorem 2

Let $F = A_1* \cdots *A_n * F_k$ be a free product of the nontrivial finite groups $A_i$ with a free group of rank $k$. The invariants $n$ and $k$ determine the number of ends of $\mathop{\mathrm{Out}}\nolimits(F)$ in the following way.

Figures (2)

  • Figure 1: The subdivided square in the proof of \ref{['compatiblepaths']}.
  • Figure 2: The simplicially subdivided polygons in the proof of \ref{['mainconnectivity']}.

Theorems & Definitions (40)

  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 1.1: cf. Lemma 2.2 of Vogtmann
  • proof
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 30 more