The Tits alternative for the automorphism group of a free product

Abstract

Let $G=G_1\ast \cdots \ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in \{1,\dots ,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtually polycyclic groups.

arXiv commentv2: Revised version: introduction rewritten, exposition improved, typos corrected

References in corpus (13)

• Relative hyperbolicity and relative quasiconvexity for countable groups
• Automorphisms of two-dimensional right-angled Artin groups
• Limit groups and groups acting freely on R^n-trees
• JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space
• Reducing systems for very small trees
• Subgroup classification in Out(F_n)
• JSJ decompositions: definitions, existence, uniqueness. II. Compatibility and acylindricity
• Actions of finitely generated groups on R-trees
• The Tits Alternative for $Out(F_n)$ II: A Kolchin Type Theorem
• Relative free splitting and free factor complexes I: Hyperbolicity
• Solvable subgroups of Out(F_n) are virtually abelian
• A short proof of Handel and Mosher's alternative for subgroups of $\text{Out}(F_N)$
• Translation lengths in Out(F_n)

Cited by in corpus (15)

• Automorphisms of graph products of groups from a geometric perspective
• Algebraic laminations for free products and arational trees
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Relative automorphism groups of right-angled Artin groups
• Irreducible laminations for IWIP Automorphisms of free products and Centralisers
• The Conjugacy Problem for $Out(F_3)$
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Outer automorphism groups of graph products: subgroups and quotients
• Automorphism groups of cyclic products of groups
• Commensurations of subgroups of $\mathrm{Out}(F_N)$
• Free representations of outer automorphism groups of free products via characteristic abelian coverings
• Extensions of hyperbolic groups have locally uniform exponential growth
• Limit trees for free group automorphisms: universality
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
• Train track maps for graphs of groups