Arithmetic and Zariski-dense subgroups in algebraic groups

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The project addresses several important problems in the investigation of arithmetic and general Zariski-dense subgroups of semisimple algebraic groups. One of the central themes in the project is the analysis of weakly commensurable Zariski-dense subgroups. The notion of weak commensurability was introduced earlier in a joint work of the PI and G. Prasad, and its analysis for arithmetic groups has led to a resolution of several problems in differential geometry dealing with length-commensurable and isospectral arithmetically defined locally symmetric spaces. In the current project, some finiteness results which were previously known only for arithmetic groups are expected to be generalized to arbitrary Zariski-dense subgroups. This work is likely to have interesting consequences for non-arithmetically defined locally symmetric spaces. It is also related to the problem in the theory of algebraic groups of to what extent an absolutely almost simple algebraic group over a field K is determined by the isomorphism classes of its maximal K-tori. This part of the project will build on the PI's recent results analyzing division algebras having the same maximal subfields. The PI also intends to continue the investigation of the congruence subgroup problem. Generally speaking, the project focuses on the analysis of a very broad class of matrix groups (Zariski-dense subgroups of semisimple algebraic groups) based on the information about the eigenvalues of their elements. This approach is fundamental in the representation theory of finite groups, but as was discovered by the PI and G. Prasad it can also be used to characterize many arithmetic groups (which are special groups whose elements are matrices with integer matrices). The goal of the project is to extend some of these results to groups much more general than arithmetic. This work has applications to the famous question ''Can one hear the shape of a drum?'' in the context of some special spaces called locally symmetric spaces.