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The simplest class of non-abelian Lie groups is (arguably) (P)SL(n, R ): the group of nxn matrices with determinant one. This is a Lie group of real rank n-1 -- the rank of a Lie group is the dimension of its maximal torus (maximal abelian subgroup). By analogy with the abelian case, we next define a lattice in SL(n) (or any Lie group, for that matter):
work with the FINITE groups SL(n, Z /p Z ), and show that a random walk on that group must become equidistributed, and then analyse the morphism from the group to the set of characteristic polynomials.
Then use “strong approximation” as the analogue of “Chinese remaindering” to lift the result to SL(n, Z ).
Amazing fact: For a Zariski dense subgroup modP is surjective except for finitely many exceptions (Matthews-Vaserstein-Weisfeiler 1987) and conversely, if modP is onto for some prime > 2 (T. Weigel, 200?), then the subgroup is Zariski-dense.
The number of exceptions in MVW is effectively computable, supposedly (I have never seen an effective bound).
It is known that the “generalized word problem” or “membership problem” is undecidable for SL(n, Z ) for n>3 (decidable for n=2, OPEN for n=3) -- reduces to Post Correspondence, since the groups contain F 2 xF 2 .
Membership problem: does a group generated by A, B, C, ..., D contain a given matrix M?