Mathematics Colloquia and Seminars

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I will start by recalling the work of van Kampen and his method of proving that e.g. the "utilities graph" does not embed in the plane (plus the higher-dimensional analogs). I will then give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, the $n$-fold product of nonabelian free groups cannot act properly discontinuously on $R^{2n-1}$. The idea is to find a utility graph (or an analog) "at infinity of the group". If there is time, I will discuss why SL_n(Z) cannot act properly discontinuously on a contractible manifold whose dimension is less than the dimension of the symmetric space SL_n(R)/SO_n.