Discrete Geometry: Growth and nonamenability in product replacement graphs

Event Date: 

Wednesday, October 16, 2013 -
2:00pm to 3:00pm

Event Location: 

  • 4607B South Hall

Event Contact: 

Michael Dougherty

Email: dougherty@math.ucsb.edu

Speaker: Anton Malyshev (UCLA)

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.