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Relative bounded cohomology on groups with contracting elements (2409.20348v2)
Published 30 Sep 2024 in math.GR and math.GT
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 $H{2}_b(G, \langle \langle gk\rangle \rangle; \mathbb{R})$ 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.