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When is the Outer Space of a free product CAT(0)?

Robert Alonzo Lyman

TL;DR

The paper investigates when the Outer Space for a free product $G=A*B*\mathbb{Z}$ can be equipped with a $\mathrm{Out}(G)$-equivariant CAT$(0)$ metric on its spine $L(G)$. It proves a targeted positive result: when $G=A*B*\mathbb{Z}$ with $A,B$ nontrivial finite, the spine $L(G)$ is 2-dimensional, admits a geometric $\mathrm{Out}(G)$-action, and supports a $\mathrm{Out}(G)$-equivariant CAT$(0)$ metric; in the special case $A=B=C_2$, $L$ is isometric to the Davis–Moussong complex of a Coxeter group. Conversely, the paper establishes a broad negative result: if the spine has dimension at least $3$ (and more generally outside a finite list of decompositions), no $\mathrm{Out}(G,\mathscr{A})$-equivariant piecewise-Euclidean or piecewise-hyperbolic CAT$(0)$ metric exists; the obstruction is detected via Gromov’s link condition, angle considerations in $2$-simplices, and end-structure arguments showing infinitely many ends. Overall, the work delineates a near-complete dichotomy: a two-dimensional CAT$(0)$ Outer Space can occur only in a very restricted family (notably $G=A*B*\mathbb{Z}$ with finite $A,B$), while higher-dimensional cases fail to admit such metrics except in special trivial decompositions. This advances understanding of geometric structures on automorphism groups of free products and their deformation-analytic spine complexes, with implications for the coarse geometry of $\mathrm{Out}(G)$ and related deformation spaces.

Abstract

Generalizing Culler and Vogtmann's Outer Space for the free group, Guirardel and Levitt construct an Outer Space for a free product of groups. We completely characterize when this space (or really its simplicial spine) supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham. In particular, provided the dimension of the spine is at least three, it is never CAT(0). Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional and does support an equivariant CAT(0) metric.

When is the Outer Space of a free product CAT(0)?

TL;DR

The paper investigates when the Outer Space for a free product can be equipped with a -equivariant CAT metric on its spine . It proves a targeted positive result: when with nontrivial finite, the spine is 2-dimensional, admits a geometric -action, and supports a -equivariant CAT metric; in the special case , is isometric to the Davis–Moussong complex of a Coxeter group. Conversely, the paper establishes a broad negative result: if the spine has dimension at least (and more generally outside a finite list of decompositions), no -equivariant piecewise-Euclidean or piecewise-hyperbolic CAT metric exists; the obstruction is detected via Gromov’s link condition, angle considerations in -simplices, and end-structure arguments showing infinitely many ends. Overall, the work delineates a near-complete dichotomy: a two-dimensional CAT Outer Space can occur only in a very restricted family (notably with finite ), while higher-dimensional cases fail to admit such metrics except in special trivial decompositions. This advances understanding of geometric structures on automorphism groups of free products and their deformation-analytic spine complexes, with implications for the coarse geometry of and related deformation spaces.

Abstract

Generalizing Culler and Vogtmann's Outer Space for the free group, Guirardel and Levitt construct an Outer Space for a free product of groups. We completely characterize when this space (or really its simplicial spine) supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham. In particular, provided the dimension of the spine is at least three, it is never CAT(0). Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional and does support an equivariant CAT(0) metric.
Paper Structure (26 sections, 14 theorems, 11 equations, 7 figures)

This paper contains 26 sections, 14 theorems, 11 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The complex
  3. Graphs of groups and maps thereof
  4. Graphs of groups.
  5. The fundamental group.
  6. Maps of graphs of groups.
  7. Deformation spaces and their spines
  8. Links in K and L
  9. Blowing up ideal edges.
  10. Links in $L$.
  11. The proof of Theorem A
  12. The case of C2 * C2 * Z
  13. Intermediate graph of groups have two expansions.
  14. Stars of minimal graphs of groups are polygons.
  15. $L$ is CAT(0).
  16. ...and 11 more sections

Key Result

Theorem 1

For any nontrivial finite groups $A$ and $B$, let $G = A*B*\mathbb{Z}$. The group $\mathop{\mathrm{Out}}\nolimits(G)$ of outer automorphisms of $G$ satisfies the following properties.

Figures (7)

  • Figure 1: The combinatorial types of graphs of groups occuring in $L(F)$.
  • Figure 2: The link of a maximal marked graph of groups.
  • Figure 3: A portion of the complex $L$.
  • Figure 4: The pieces the complex $L$ is tiled by.
  • Figure 5: A $2$-simplex in $L$, two of whose angles must but cannot be at least $\frac{\pi}{2}$.
  • ...and 2 more figures

Theorems & Definitions (33)

  • Theorem 1
  • proof
  • Theorem 2
  • Definition 2.1: Quotient graph of groups
  • Definition 2.2
  • Definition 2.3: Map of graphs of groups
  • Definition 2.4: Homotopy of maps of graphs of groups
  • Definition 2.5
  • Definition 2.6: Collapsible edges, collapses and their homotopy inverses, closed cones
  • Definition 2.7: The spine of $\mathscr{PT}(\mathbb{G})$
  • ...and 23 more