Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces

Summary

The article contains a survey of our results on length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semisimple algebraic groups. We discuss some of the techniques involved in this work (in particular, the existence of generic tori in semisimple algebraic groups over finitely generated fields and of generic elements in finitely generated Zariski-dense subgroups) and some open problems.

This article contains an exposition of recent results on isospectral and length-commensurable locally symmetric spaces associated with simple real algebraic groups [Prasad and Rapinchuk 2009; 2013] and related problems in the theory of semisimple algebraic groups [Garibaldi 2012; Garibaldi and Rapinchuk 2013; Prasad and Rapinchuk 2010a]. One of the goals of [Prasad and Rapinchuk 2009] was to study the problem beautifully formulated by Mark Kac in [1966] as “Can one hear the shape of a drum?” for the quotients of symmetric spaces of the groups of real points of absolutely simple real algebraic groups by cocompact arithmetic subgroups. A precise mathematical formulation of Kac’s question is whether two compact Riemannian manifolds which are isospectral (i.e., have equal spectra—eigenvalues and multiplicities—for the Laplace–Beltrami operator) are necessarily isometric. In general, the answer to this question is in the negative as was shown by John Milnor already in [1964] by constructing two nonisometric isospectral flat tori of dimension 16. Later M.-F.Vignéras [1980] used arithmetic properties of quaternion algebras to produce examples of arithmetically defined isospectral, but not isometric, Riemann surfaces. On the other hand, T. Sunada [1985], inspired by a construction of nonisomorphic number fields with the same Dedekind zeta-function, proposed a general and basically purely group-theoretic method of producing nonisometric isospectral Riemannian manifolds which has since then been used in various ways. It is important to note, however, that the nonisometric isospectral manifolds constructed by Vignéras and Sunada are commensurable, that is, have a common finite- sheeted cover. This suggests that one should probably settle for the following weaker version of Kac’s original question: Are any two isospectral compact Riemannian manifolds necessarily commensurable? The answer to this modified question is still negative in the general case: Lubotzky, Samuels and Vishne [Lubotzky et al. 2006], using the Langlands correspondence, have constructed examples of noncommensurable isospectral locally symmetric spaces associated with absolutely simple real groups of type A_n (see Problem 10.7). Nevertheless, it turned out that the answer is actually in the affirmative for several classes of locally symmetric spaces. Prior to our paper [Prasad and Rapinchuk 2009], this was known to be the case only for the following two classes: arithmetically defined Riemann surfaces [Reid 1992] and arithmetically defined hyperbolic 3-manifolds [Chinburg et al. 2008].