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4:00 pm Monday, March 18, 2013 Topology Seminar: SL(n, Q) has no volume-preserving actions on (n − 1)-dimensional compact manifoldsby Dave Witte-Morris (Lethbridge) in HB 427- As part of the "Zimmer program," numerous authors have studied volume-preserving actions of the group SL(n,A) on compact manifolds, where A is either the ring Z of integers or the field R of real numbers. On the other hand, very little seems to be known about the intermediate case where A is the field Q of rational numbers. As a first step in this direction, we show that SL(n,Q) has no nontrivial, C-infinity, volume-preserving action on any compact manifold of dimension strictly less than n. The proof has two main ingredients: a theorem of Zimmer tells us that the action of any "S-arithmetic" subgroup must extend (a.e.) to a measurable action of its profinite completion, and the Congruence Subgroup Property provides a very nice description of this profinite completion. This is joint work with Robert J. Zimmer of the University of Chicago.
Submitted by andyp@rice.edu |