Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings

Abstract

We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast \cdots \ast G_k\ast F$, where $F$ denotes a finitely generated free group, we identify the Gromov boundary of the graph of relative cyclic splittings with the space of equivalence classes of $\mathcal{Z}$-averse trees in the boundary of the corresponding outer space. A tree is \emph{$\mathcal{Z}$-averse} if it is not compatible with any tree $T^{\prime }$, that is itself compatible with a relative cyclic splitting. Two $\mathcal{Z}$-averse trees are \emph{equivalent} if they are both compatible with a common tree in the boundary of the corresponding outer space. We give a similar description of the Gromov boundary of the graph of maximally-cyclic splittings.

v3: Final version, incorporating the referee's suggestions ; to appear in the Journal of Topology

References in corpus (3)

• The boundary of the complex of free factors
• On hyperbolicity of free splitting and free factor complexes
• Subfactor projections

Cited by in corpus (8)

• 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 hyperbolicity for automorphisms of free products
• Spectral theorems for random walks on mapping class groups and $\text{Out}(F_N)$
• On the topological dimension of the Gromov boundaries of some hyperbolic $Out(F_N)$-graphs
• Loxodromic elements in the cyclic splitting complex and their centralizers
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$