Trees of multidisks as Gromov boundaries of amalgamated products

TL;DR

This paper characterizes the Gromov boundary of certain free products of hyperbolic groups amalgamated over a common subgroup, using trees of multidisks and topological sphere structures.

Contribution

It provides a description of the Gromov boundary for amalgamated free products of hyperbolic groups with specific boundary topologies, under technical assumptions.

Findings

Gromov boundary is homeomorphic to a tree of multidisks.

Boundary of each hyperbolic group resembles a densely punctured sphere.

Results extend understanding of boundaries in complex group amalgamations.

Abstract

We describe, under some additional technical assumptions, the Gromov boundary of the free product of several $G_i$'s amalgamated wrt. $H$, where $G_i$ are hyperbolic groups with boundary homeomorphic to a densely punctured $n$-sphere, and $H$ is their common subgroup corresponding to a peripheral sphere in each of the $\partial G_i$'s.