Geometry & Topology Seminar: Torsion Invariants for Hecke Correspondences on Compact Hyperbolic Spaces

Abstract: We begin with the Reidemeister (R-) torsion, a combinatorial invariant of manifolds with flat vector bundles, and its analytic counterpart defined via Laplacian determinants. The  Cheeger–Müller theorem identifies these two notions of torsion in the acyclic case. We then turn to the Ruelle zeta function, a dynamical invariant built from the closed geodesics of the hyperbolic manifold, and discuss Fried’s conjecture — which relates its value at zero to analytic torsion. In this setting, Hecke correspondences introduce an arithmetic structure, allowing us to refine these torsion invariants and zeta functions along Hecke symmetries. The talk will survey both classical results and recent developments, including extensions of  Fried’s conjecture under Hecke correspondence.

Host: Yanli Song