Mathematics Ph.D. Dissertations

Title

Properties of groups acting on Twin-Trees and Chabauty space

Date of Award

2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Rieuwert Blok (Advisor)

Second Advisor

Lee Nickoson (Other)

Third Advisor

Mihai Staic (Committee Member)

Fourth Advisor

Xiangdong Xie (Committee Member)

Abstract

In this dissertation, we study groups that act on twin trees. A twin tree consists of a pair of (infinite) simplicial trees (X+, X-) that are ``twinned" by means of a co-distance function δ*, which assigns a non-negative integer to pairs of vertices from each tree. If n=δ*(x+, y-) for vertices x+ in X+ and y- in X-, then we think of x+ and y- as having distance ∞ - n. An example of a twin tree is Τ=(Τ+, Τ-, δ*), where Τ+ and Τ- are the associated Bruhat-Tits trees arising from two different discrete valuations on the field k(t).

A group G acts on a twin tree X=(X+, X-, δ*) if it acts on each tree X+, X- and preserves the co-distance function. For the twin tree Τ arising from discrete valuations on k(t), the group GL(2,k[t,t^-1]) naturally acts on the twinning. The subgroup GL(2,k[t]) stabilizes a vertex of the the tree Τ+. The action of GL(2,k[t]) on Τ- yields a fundamental domain an infinite ray, and from this action one obtains Nagao's Theorem. In this work, we investigate the fundamental domains for subgroups G < GL(2,k[t,t^-1]) that stabilize subtrees of the tree Τ+.

For a general group G acting on a twin-tree, we consider its space of closed subgroups C(G), called the Chabauty space. By constructing a left-invariant metric on the underlying automorphism group of the twin-tree, one can endow C(G) with a metric as well. Using this, we study the distance between vertex stabilizer subgroups in G. This will hopeful lead to future work generalizing the special case of Τ and GL(2,k[t,t^-1]).

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