We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric spaces.

/pdf/weakly-commensurable-arithmetic-groups-and-isospectral-3q5rxiq0kc.pdf

Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

Isospectrality and commensurability of arithmetic hyperbolic $2$- and $3$-manifolds

We show that if M is an arithmetic hyperbolic 3-manifold, the set QL(M) of all rational multiples of lengths of closed geodesics of M both determines and is determined by the commensurability class of M. This implies that the spectrum of the Laplace operator of M determines the commensurability class of M. We also show that the zeta function of a number field with exactly one complex place determines the isomorphism class of the number field

/pdf/geodesics-and-commensurability-classes-of-arithmetic-4ml06exuoy.pdf

Geodesics and commensurability classes of arithmetic hyperbolic $3$-manifolds

Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces

/pdf/zeta-functions-of-selberg-s-type-associated-with-homogeneous-34lp2b2uv2.pdf

Ramesh Gangolli

Papers

The length spectra of some compact manifolds of negative curvature