Dead end depth

Seminar: 
Group Actions and Dynamics
Event time: 
Monday, September 27, 2004 - 12:30pm to Sunday, September 26, 2004 - 8:00pm
Location: 
431 DL
Speaker: 
Tim Riley
Speaker affiliation: 
Yale University
Event description: 

I will exhibit groups with Cayley graphs in which balls and geodesics exhibit exotic behaviour.

The dead end depth of an element g of a group G, with respect to a generating set A, is the distance from g to the complement of the radius d(1,g) closed ball, in the word metric d defined with respect to A. The lamplighter group, with standard finite generating sets, contains elements of arbitrarily large dead end depth. But the lamplighter group is not finitely presentable, and it has been asked (first by O. Bogopolskii) whether there is a finitely presentable group with a finite generating set with respect to which there is no upper bound on the dead end depth. I will exhibit such a group (work with S. Cleary).

A further natural question (asked by J. Taback) is whether the property of having unbounded dead end depth is a quasi-isometry invariant, or indeed whether it is even a group invariant. I will answer this in the negative by giving a group that has unbounded dead end depth with respect to one finite generating set, but dead end depth identically zero with respect to another.