Computing the Cech cohomology of decomposition spaces

TL;DR

This paper introduces an algorithm to compute the cech cohomology of decomposition spaces associated with line patterns in free groups, addressing a question about boundaries of hyperbolic groups.

Contribution

It provides the first algorithmic method to determine the cech cohomology of these specific decomposition spaces as an $F$-module.

Findings

Algorithm successfully computes cech cohomology presentations

Answers a relative version of Epstein's boundary question

Advances understanding of hyperbolic group boundaries

Abstract

A line pattern in a free group $F$ is defined by a malnormal collection of cyclic subgroups. Otal defined a decomposition space $\mathcal{D}$ associated to a line pattern. We provide an algorithm that computes a presentation for the \v{C}ech cohomology of $\mathcal{D}$, thought of as a $F$-module. This answers a relative version of a question of Epstein about boundaries of hyperbolic groups.