New properties of arithmetic groups

V. P. Platonov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 5, pp. 951-975

Résumé

New substantial results including the solutions of a number of fundamental problems have been obtained in the last decade or so: the first and rather unexpected examples of arithmetic groups with finite extensions that are not arithmetic were constructed; a criterion for arithmeticity of such extensions was found; deep rigidity theorems were proved for arithmetic subgroups of algebraic groups with radical; a theorem on the finiteness of the number of conjugacy classes of finite subgroups in finite extensions of arithmetic groups was proved, leading to numerous applications, in particular, this theorem made it possible to solve the Borel–Serre problem (1964) on the finiteness of the first cohomology of finite groups with coefficients in an arithmetic group; the problem posed more than 30 years ago on the existence of finitely generated integral linear groups that have infinitely many conjugacy classes of finite subgroups was solved; the arithmeticity question for solvable groups was settled. Similar problems were also solved for lattices in Lie groups with finitely many connected components. This paper is a survey of these results. Bibliography: 27 titles.

Détail

Keywords: arithmetic group, rigidity theorems, arithmeticity criterion, problem of conjugacy of finite subgroups, lattices in Lie groups.

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