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#### Author ORCID Identifier

0000-0003-4619-1087

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Degree Program

Mathematics

#### Year Degree Awarded

2019

#### Month Degree Awarded

September

#### First Advisor

Paul Gunnells

#### Subject Categories

Algebra | Geometry and Topology | Number Theory

#### Abstract

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main result gives an explicit formula for the homology of the quotient space Gamma(g)\X2 (Theorem 1.5.1). We also give a complete description of SLn(A)\X3 (Theorem 2.2.5), and explicitly compute the stabiliser groups of the cells therein (Theorem 2.2.3). Using this data we derive information about the topology and simplicial structure of the quotient space Gamma(g)\X3 (Theorem 2.3.7), and explicitly compute the homology groups (Theorem 2.4.4). We also define an appropriate generalisation of unimodular symbols for SL3(F) (Definition 3.3.1), and prove that a continued fraction type algorithm exists (in the sense of [AR79]), thus showing any modular symbol can be written a sum of unimodular symbols (Theorem 3.3.2).

#### DOI

https://doi.org/10.7275/y00y-js17

#### Recommended Citation

Bates, Matthew, "Bruhat-Tits Buildings and a Characteristic p Unimodular Symbol Algorithm" (2019). *Doctoral Dissertations*. 1689.

https://doi.org/10.7275/y00y-js17
https://scholarworks.umass.edu/dissertations_2/1689