On spectral measures in semidirect products and the decidability of the zero divisor problem

Łukasz Grabowski (U Oxford)

Wednesday 27th February, 2013 16:00-17:00 204


I will start the talk by outlining a general method for computing spectral meaures of elements of group rings of groups which are semidirect products in a non-trivial way. The method has had several applications so far - showing that every non-negative real number is an l2-Betti number, showing that 0 can be a Novikov-Shubin invariant, and showing existence of random walks on groups with singular continuous spectral measure. Depending on the interests of the audience I can explain some of these results. Otheriwse I'll talk about my recent result which uses the same method - denote by R the integral group ring of the group G^4, where G is the so-called lamplighter group. Then, there is no algorithm to decide if a given element of R is a 0-divisor. I will briefly discuss relation of this "zero divisor problem" to other decision problems studied in group theory, and I'll finish by mentioning other decision problems concering group rings where I have no idea if there exist algortihms which solve these problems.

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