On super-rigidity of Gromov’s random monster group

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发表日期：

2024年11月14日

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Kajal Das

摘要

In this article, we show super-rigidity of Gromov’s random monster group. It is known from a paper of Assaf Naor and Lior Silberman that any homomorphic image of Gromov’s random monster group into a linear group is finite. It can be also derived from the previously known results that the same result is true for a- $L^p$ $L^{p}$ -menable groups and K-amenable groups. We extend these results and prove that any morphism $\phi _\alpha $ $ϕ_{α}$ from Gromov’s random monster group $\Gamma _\alpha $ $Γ_{α}$ to a countable discrete group G has finite image for almost all $\alpha $ $α$ , where G is any of the following types of groups: mapping class group $MCG(S_{g,b})$ $M C G (S_{g, b})$ , braid group $B_n$ $B_{n}$ , outer automorphism group of a free group $Out(F_N)$ $O u t (F_{N})$ , automorphism group of a free group $Aut(F_N)$ $A u t (F_{N})$ and hierarchically hyperbolic group. For acylindrically hyperbolic groups, we deduce that the homomorphic image is absolutely elliptic. We introduce another property called hereditary super-rigidity, which is the property of super-rigidity for all finite-index sub-groups. It immediately follows from the literature that $\Gamma _\alpha $ $Γ_{α}$ has hereditary super-rigidity with respect to an a- $L^p$ $L^{p}$ -menable group or a K-amenable group for a.e. $\alpha $ $α$ . In this article, we establish a stability result for the groups with respect to which $\Gamma _\alpha $ $Γ_{α}$ has super-rigidity and hereditary super-rigidity.