##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Anton Malyshev

#### UCLA

## Growth and nonamenability in product replacement graphs

##### Abstract:

The product replacement graph (PRG) of a group G is the set of generating k-tuples of G, with edges corresponding to Nielsen moves. It is conjectured that PRGs of infinite groups are nonamenable. We verify that PRGs have exponential growth when G has polynomial growth or exponential growth, and show that this also holds for a group of intermediate growth: the Grigorchuk group. We also provide some sufficient conditions for nonamenability of the PRG, which cover elementary amenable groups, linear groups, and hyperbolic groups.

Host: Efim Zelmanov

### December 2, 2013

### 2:00 PM

### AP&M 7218

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