Colloquium| Institute of Mathematical Sciences
Time: Wednesday, Apirl 17th, 2024 , 13:15-14:15
Location：IMS, RS408
Speaker: Zhenguo Huangfu, Peking University
Abstract: Let $\Gamma$ be a non-elementary hyperbolic group and let $H$ be a quasi-convex subgroup of $\Gamma$. We prove that $H$ has infinite index in $\Gamma$ if and only if the relative second bounded cohomology $\mathbf{H}^{2}_b(\Gamma, H; \mathbb{R})$ is infinite dimensional. We follow the approach of D. Epstein and K. Fujiwara through which they proved that $\mathbf{H}^{2}_b(\Gamma; \mathbb{R})$ is infinite dimensional if $\Gamma$ is a non-elementary hyperbolic group. We use the same infinite sequence of quasimorphisms such that their coboundaries are $\mathbb{R}$-linearly independent in $\mathbf{H}^{2}_b(\Gamma; \mathbb{R})$, which was explicitly constructed by M. Bestivina and K. Fujiwara for absolute second bounded cohomology of hyperbolic groups. With a careful analysis on the Gromov boundary of $\Gamma$ and of $H$, we are able to show that those quasimorphisms have an additional property of vanishing on $H$. Therefore their coboundaries represent linearly independent coclasses in $\mathbf{H}^{2}_b(\Gamma, H; \mathbb{R})$. Our results generalize a theorem of C. Pagliantini and P. Rolli which states that a finitely generated subgroup $H$ has infinite index in a rank $n$ free group $F_n$ if and only if $\mathbf{H}^{2}_b(F_n, H; \mathbb{R})$ is infinite dimensional. This is a joint work with Professor Wenyuan Yang(Peking University).