Beyond Serre's "Trees" in two directions: $\Lambda$--trees and products of trees

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 $\Lambda$-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.