Cohomology of Fuchsian Groups and Non-Euclidean Crystallographic Groups

TL;DR

This paper computes the cohomology groups for geometrically finite 2D non-Euclidean crystallographic groups, specifically Fuchsian groups, and determines their ring structures, advancing understanding of their algebraic topology.

Contribution

It provides explicit cohomology calculations and ring structures for Fuchsian and NEC groups, which were previously not fully characterized.

Findings

Cohomology groups are explicitly computed for all geometrically finite NEC groups.

Ring structures of cohomology are determined for Fuchsian groups.

Results enhance the algebraic understanding of non-Euclidean crystallographic groups.

Abstract

For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.