Arithmetic Groups

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We plan to study arithmetic groups for their own sake and for theirapplications to other branches of mathematics and computer science.Two central themes are the subgroup growth of arithmetic lattices and thecombinatorics of the simplicial complexes obtained by taking quotients ofBruhat-Tits buildings modulo arithmetic groups.Arithmetic groups are at the crossroads of several central areas ofmathematics: Lie groups, number theory, geometry and combinatorics.Their study can be carried out using a wealth of different methods andshed light on different directions. A previous work has led to theconstruction of Ramanujan graphs which turn out to have remarkableapplications to computer science and combinatorics. One of the goals ofthis research will be to study higher dimensional analogues with thehope of getting deeper applications.