Examples of Homologically Wide Actions

Abstract

AbstractAn action of a finite groupGon a manifoldMis homologically wide if the first homology of the manifold contains the regular representation of the group. In this chapter, we study this notion independently of the rest of this monograph. We first study the case whereMis a surface andGacts freely, using the Lefschetz fixed point formula. We then recall a result of Broughton on the homology representation that allows us to deal with the case of a general action on a Riemann surface. After this, we switch to higher dimensional manifolds. We first recall Curtis’s theory of the virtual Lefschetz characters, and use results of Cooper and Long to construct examples and counterexamples to homological wideness in all dimensions ≥ 3. Finally, we study locally symmetric spaces. Property (T) implies that only the case of rank 1 is interesting, where we make some remarks on the relation to automorphic representations and Mostow rigidity. Finally, we study an example of using torsion homology: Mednykh’s explicit computation of the homology representation of the Seifert-Weber dodecahedral space, which we identify with a known modular representation through Brauer characters.