A generalization of the distortion function and the asymptotic geometry of subgroups
Jarnevic, Andy
0009-0009-8521-0552
:
2023-06-14
Abstract
This thesis introduces and studies a natural generalization of the distortion
function that applies to not necessarily finitely generated subgroups of finitely
generated groups. We begin by computing this function in several natural cases,
and provides an example of a group with an uncountable collection of incom-
parably distorted subgroups.We then show that when we restrict this function
to the case of finitely generated subgroups H of finitely generated groups G,
the generalized distortion function characterizes when a natural subspace of the
asymptotic cone of G corresponding to H is connected. We denote this subspace
by Coneω
G(H) and show that the ordinary distortion function is not sufficient to
detect this subspace’s connectedness. We then study the convexity properties
of Coneω
G(H). We show that a subgroup H of a finitely generated group G
is strongly quasi-convex if and only if Coneω
G(H) satisfies a natural convexity
property in Coneω (G). G acts on Coneω (G) in a natural way. We show that the
stabilizer of Coneω
G(G) is the same as the commensurator of H in G whenever
H is strongly quasi-convex in G. We conclude by providing several applications
of this result to groups with Morse elements.