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Title: Asymptotics in conjugacy classes for free groups
Author: Kenison, George
ISNI:       0000 0004 6496 7886
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2017
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In this thesis we consider asymptotic counts of words in free groups. In particular, we establish results when we restrict the group elements to a non-trivial conjugacy class. We study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Our results require an irrationality condition on the edge lengths. We obtain an asymptotic formula for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most T. Under a mild extra assumption we also obtain a polynomial error term. Related to the above orbit counting result for metric trees, we also consider the spatial distribution of the lattice points of a given conjugacy class in the universal covering tree. We show that the lattice points of a fixed conjugacy class are asymptotically spatially distributed according to a Patterson-Sullivan measure supported on the boundary of the universal cover. For a class of functions on a free group with suitable symbolic properties we establish an asymptotic average and, subject to an appropriate normalisation, a central limit theorem when the elements of the free group are restricted to a non-trivial conjugacy class.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics