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Properties of groups acting on Twin-Trees and Chabauty space

Kelvey, Robert J, Kelvey
http://orcid.org/0000-0003-0970-2706
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1479423366082688

Abstract Details

2016, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics.
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]).
Rieuwert Blok (Advisor)
Lee Nickoson (Other)
Mihai Staic (Committee Member)
Xiangdong Xie (Committee Member)
215 p.
Mathematics
trees; twin-trees; group theory; geometric group theory; buildings; twin-buildings; Chabauty space; Bruhat-Tits tree; Bass-Serre theory; fundamental domains; lattices; Nagao lattices

Recommended Citations

Citations

  • Kelvey, Kelvey, R. J. (2016). Properties of groups acting on Twin-Trees and Chabauty space [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1479423366082688

    APA Style (7th edition)

  • Kelvey, Kelvey, Robert. Properties of groups acting on Twin-Trees and Chabauty space. 2016. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1479423366082688.

    MLA Style (8th edition)

  • Kelvey, Kelvey, Robert. "Properties of groups acting on Twin-Trees and Chabauty space." Doctoral dissertation, Bowling Green State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1479423366082688

    Chicago Manual of Style (17th edition)

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This open access ETD is published by Bowling Green State University and OhioLINK.