## 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]).

#### Recommended Citation

Kelvey, Robert J., "Properties of groups acting on Twin-Trees and Chabauty space" (2016). *Mathematics Ph.D. Dissertations*. 11.

https://scholarworks.bgsu.edu/math_diss/11