London Algebra Colloquium abstract
Generalization of the word problem and products of lamplighters
Lukasz Grabowski (Imperial), 2nd June 2011
Abstract
In the talk I will introduce the following generalization of
the word problem for groups, which I will call the zero-divisor
problem. Given a finitely presented group G and an element T from the
integral group ring ZG, given in terms of the generators of G, decide
whether T is a zero-divisor or not. I’ll describe two
results. First, an observation that decidable word problem together
with soficity and the Atiyah conjecture implies decidability of the
zero-divisor problem. Second, I will show that there exist groups with
decidable word problem but undecidable zero-divisor problem. As
examples one can take products of the standard lamplighter group. The
main tool is going to be “embedding a Turing machine into a group
ring”, which is described in more detail in my PhD thesis.