Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups

An important cohomological property of a countable group Gamma concerns its duality property of its homology and cohomology groups with coefficients. In this paper, we review the duality property of arithmetic subgroups of linear algebraic groups and the identification of the dualizing module with the Steinberg representation (or module) of the arithmetic subgroups, discuss duality property of the mapping class groups of surfaces (for example, they are duality groups but not Poincare duality groups) and also the identification of the dualizing module with a generalized Steinberg module, and prove that the outer automorphism group Out(F-n) of free groups F-n is also a duality group but not a Poincare duality group. Based on the discussions in this paper, we also propose a conjecture that the outer automorphism group of a duality group is a virtual duality group. Hope that the discussions in this paper will add to the list of amazing applications to many different fields of the Steinberg representations.