Relative bounded cohomology on groups with contracting elements

TL;DR

The paper investigates how the infinite index of Morse subgroups inside a group $G$ acting on a space with contracting elements is reflected in the relative second bounded cohomology $H^2_b(G,\{H_i\}_{i=1}^n;\\mathbb{R})$, proving it is infinite-dimensional in this setting. It develops a two-pronged approach: (i) construct a projection complex on which $G$ acts so that all $H_i$ act elliptically, and (ii) adapt Epstein–Fujiwara quasimorphisms to produce continuum-many relative 2-cocycles vanishing on the $H_i$, ensuring the relative cohomology has continuum dimension. The authors further extend the framework to normal closures of contracting elements via rotation-family cone-offs, showing the corresponding relative cohomology $H^2_b(G, \,\langle\langle g^k\rangle\rangle;\\mathbb{R})$ also has continuum dimension for suitable $k$. Overall, the results generalize Pagliantini–Rolli for finite-ractor free groups and advance understanding of the relative bounded cohomology in groups with contracting dynamics and Morse subgroups.

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.

Figures (8)

• Figure 1: $[x,y]\subset N_{\delta}([x,z]\cup [y,z])$
• Figure 2: $Y$ is $C$-contracting
• Figure 3: $\gamma=q_1p_1q_2p_2q_3p_3q_4$ is a $(D,\tau)$-admissible path
• Figure 4: $\gamma=p_0q_1p_1q_2p_2$ is a $(D,\tau)$-admissible path and $\alpha$$\epsilon$-fellow travels $\gamma$
• Figure 5: Each blue line is a translate of $\mathrm{Ax}(g)$; $\mathcal{F}_K[U,V]\cup \{U,V\}=\{U<U_1<U_2<V\}$ is a standard path in $\mathcal{P}_K(\mathcal{F})$; the dashed line represents the closest point projection between $U$ and $U_1$; the red line is a lifted standard path from $u\in U$ to $v\in V$ in $X$

Theorems & Definitions (94)

• Theorem 1.1
• Remark 1.2
• Proposition 1.3: Proposition \ref{['Prop: Summary']}
• Corollary 1.4: Corollary \ref{['MainCor']}
• Proposition 1.6: Proposition \ref{['Prop: NormalClosure']}
• Remark 1.7
• Lemma 2.1: Morse Lemma
• Lemma 2.2
• proof
• Definition 2.3: Contracting Subset