74 research outputs found
Thompson's group F is not almost convex
We show that Thompson's group F does not satisfy Cannon's almost convexity
condition AC(n) for any integer n in the standard finite two generator
presentation. To accomplish this, we construct a family of pairs of elements at
distance n from the identity and distance 2 from each other, which are not
connected by a path lying inside the n-ball of length less than k for
increasingly large k. Our techniques rely upon Fordham's method for calculating
the length of a word in F and upon an analysis of the generators' geometric
actions on the tree pair diagrams representing elements of F.Comment: 19 pages, 7 figure
Metric Properties of Diestel-Leader Groups
In this paper we investigate metric properties of the groups
whose Cayley graphs are the Diestel-Leader graphs with respect to a
given generating set . These groups provide a geometric generalization
of the family of lamplighter groups, whose Cayley graphs with respect to a
certain generating set are the Diestel-Leader graphs . Bartholdi,
Neuhauser and Woess in \cite{BNW} show that for , is of
type but not . We show below that these groups have dead end
elements of arbitrary depth with respect to the generating set , as
well as infinitely many cone types and hence no regular language of geodesics.
These results are proven using a combinatorial formula to compute the word
length of group elements with respect to which is also proven in the
paper and relies on the geometry of the Diestel-Leader graphs.Comment: 19 page
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