Relative bounded cohomology on groups with contracting elements

TL;DR

This paper establishes a connection between the infinite index of Morse subgroups in groups with contracting elements and the infinite-dimensionality of their relative second bounded cohomology, generalizing previous results.

Contribution

It proves that infinite index Morse subgroups correspond to infinite-dimensional relative second bounded cohomology, extending known theorems to broader classes of groups.

Findings

Infinite index Morse subgroups imply infinite-dimensional relative second bounded cohomology.

Existence of contracting elements ensures infinite-dimensional cohomology for certain subgroups.

Generalizes Pagliantini-Rolli's theorem for free groups to groups with contracting elements.

Abstract

Let $G$ be a countable group acting properly on a metric space with contracting elements and $\{H_i:1\le i\le n\}$ be a finite collection of Morse subgroups in $G$. We prove that each $H_i$ has infinite index in $G$ if and only if the relative second bounded cohomology $H^{2}_b(G, \{H_i\}_{i=1}^n; \mathbb{R})$ is infinite-dimensional. In addition, we also prove that for any contracting element $g$, there exists $k>0$ such that $H^{2}_b(G, \langle \langle g^k\rangle \rangle; \mathbb{R})$ is infinite-dimensional. Our results generalize a theorem of Pagliantini-Rolli for finite-rank free groups and yield new results on the (relative) second bounded cohomology of groups.