Beyond Serre's "Trees" in two directions: $\Lambda$--trees and products of trees
Olga Kharlampovich, Alina Vdovina

TL;DR
This paper surveys generalizations of Serre's theory of groups acting on trees, focusing on isometric actions on $$-trees and lattices in products of trees, with applications to arithmetic groups.
Contribution
It introduces and discusses the extension of Serre's tree theory to $$-trees and products of trees, highlighting new results on arithmetic groups acting on these structures.
Findings
Development of the theory of $$-trees and their isometric group actions.
Results on lattices in products of trees, especially for arithmetic groups.
Insights into the structure of groups acting on generalized tree-like spaces.
Abstract
Serre in "Trees" laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat--Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from "Trees": the theory of isometric group actions on -trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Graph Theory Research · Rough Sets and Fuzzy Logic
