Lattices and simple groups in trees and buildings : constructions and classifications
(2018)
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- Authors
- Supervisors
- Caprace, Pierre-Emmanuel
- Abstract
- This thesis is devoted to the study of groups acting on Euclidean buildings of dimension 1 and 2. We construct lattices in products of trees and in triangle buildings and investigate non-discrete groups appearing alongside them. The first chapter focuses on trees whose vertices have valency at least 6, and provides a full classification of boundary 2-transitive automorphism groups of such trees whose local action at each vertex contains the alternating group. We obtain a new countable family of groups, all containing a simple subgroup of index at most 8. These non-discrete groups acting on trees also appear in the next chapter as projections of discrete groups (in fact lattices) acting on products of trees. Our first main achievement in this second chapter is the design of an algorithm that computes the two projections of a group acting regularly on the vertices of a product of two trees (under suitable local conditions). In the same context, we also construct new lattices with four orbits of vertices, which are simple and admit a very concise finite presentation. A precise objective is achieved in the third chapter: constructing a triangle building admitting a cocompact lattice and whose local projective plane at each vertex is non-Desarguesian. This solves a problem asked by W. Kantor in 1986. The last chapter aims at finding the weakest possible transitivity assumptions on an exotic triangle building ensuring that its full automorphism group is discrete. Its main results provide a partial answer to a question asked by T. Steger in 2007.
- Affiliations
Citations
Radu, N. (2018). Lattices and simple groups in trees and buildings : constructions and classifications. https://hdl.handle.net/2078.5/46099