Acylindrical Hyperbolicity of Mapping Class Groups and the Bounded Cohomology of High-Rank Lattices

This monograph investigates the sharp cohomological dichotomy between Mapping Class Groups of finite-type surfaces and lattices in higher-rank semisimple Lie groups. We analyze the acylindrical hyperbolicity of the Mapping Class Group Mod(g), derived from its action on the curve complex C(g), and demonstrate how this geometric property implies the infinite dimensionality of the second bounded cohomology group H2b(Mod(g); R). In contrast, we examine the rigidity phenomena characterizing high-rank lattices < G, specifically the Burger-Monod rigidity theorems which establish the injectivity of the comparison map H2b(; R) H2(; R). By juxtaposing the weak proper discontinuity (WPD) of elements in Mod(g) against the property (T) and bounded generation features of high-rank lattices, we elucidate the structural boundaries between rank-one behavior and higher-rank rigidity through the lens of bounded cohomology.