376 research outputs found
Distortion of wreath products in some finitely presented groups
Wreath products such as Z wr Z are not finitely-presentable yet can occur as
subgroups of finitely presented groups. Here we compute the distortion of Z wr
Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's
metabelian group G.
We find that Z wr Z is undistorted in F but is at least exponentially
distorted in G.Comment: 9 pages, 5 figure
Bounding right-arm rotation distances
Rotation distance measures the difference in shape between binary trees of
the same size by counting the minimum number of rotations needed to transform
one tree to the other. We describe several types of rotation distance where
restrictions are put on the locations where rotations are permitted, and
provide upper bounds on distances between trees with a fixed number of nodes
with respect to several families of these restrictions. These bounds are sharp
in a certain asymptotic sense and are obtained by relating each restricted
rotation distance to the word length of elements of Thompson's group F with
respect to different generating sets, including both finite and infinite
generating sets.Comment: 30 pages, 11 figures. This revised version corrects some typos and
has some clearer proofs of the results for the lower bounds and better
figure
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
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