On super-rigidity of Gromov's random monster group (Q6647734)

Informally, a Gromov monster is a finitely generated group which contains, in a suitable sense, a sequence of expander graphs in its Cayley graph. These Cayley graphs will admit no uniform embedding into a Hilbert space and hence these groups serve as an important source of counterexamples.\N\NIn the article under review, the author shows super-rigidity of Gromov's random monster group. It is known that any homomorphic image of Gromov's random monster group into a linear group is finite (see [\textit{A. Naor} and \textit{L. Silberman }, Compos. Math. 147, No. 5, 1546--1572 (2011; Zbl 1267.20057)]) and the same result is true for a-\(L^{p}\)-menable groups and \(K\)-amenable groups. The author extends these results and prove that any morphism \(\phi_{\alpha}\) from Gromov's random monster group \(\Gamma_{\alpha}\) to a countable discrete group \(G\) has finite image for almost all \(\alpha\), where \(G\) is any of the following types of groups: mapping class group \(MCG(S_{g,b})\), braid group \(B_{n}\), automorphism group and outer automorphism group of a free group \(F_{N}\) and hierarchically hyperbolic group. For acylindrically hyperbolic groups, he deduces that the homomorphic image is absolutely elliptic.\N\NThe author introduces another property called hereditary super-rigidity, which is the property of super-rigidity for all finite-index sub-groups. It immediately follows from from known results that \(\Gamma_{\alpha}\) has hereditary super-rigidity with respect to an a-\(L^{p}\)-menable group or a \(K\)-amenable group for almost every \(\alpha\). In this paper, the author establishes a stability result for the groups with respect to which \(\Gamma_{\alpha}\) has super-rigidity and hereditary super-rigidity.