Topology & Geometric Group Theory Seminar

Fall 2010

1:30 – 2:30, Malott 253

Tuesday, October 5

Jim Davis, Indiana University

Equivariant rigidity, Smith theory, and actions on tori

I will state the problem of equivariant rigidity for a discrete group G. A complete analysis will be made for the crystallographic group Gn given by the semidirect product of Z/2 acting on Zn by multiplication by -1. I will discuss the theorem that Gn satisfies equivariant rigidity when n is congruent to 0 or 1 modulo 4 and the constructions of counterexamples when n > 3 is congruent to 2 or 3 modulo 4.

Equivalently, one classifies involutions on tori which induce multiplication by -1 on the first homology.

Ingredients are Smith theory, surgery theory, and the Farrell–Jones conjecture. A key aspect of the application of Smith theory is the use of a theorem from point-set topology that a connected complete metric space is path-connected.

This is joint work with Qayum Khan and Frank Connolly.

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