Directions and scale-multiplicative semigroups in restricted Burger-Mozes groups

TL;DR

This paper investigates the properties of restricted Burger-Mozes groups, focusing on the scale function, space of directions, and scale-multiplicative semigroups, revealing their connections to the groups' intrinsic actions and structures.

Contribution

It provides a formula for the scale function, links the space of directions to group actions on trees and cube complexes, and constructs maximal scale-multiplicative semigroups.

Findings

Derived a formula for the scale function.

Connected the space of directions to actions on trees and CAT(0) cube complexes.

Constructed maximal scale-multiplicative semigroups.

Abstract

We study the scale function, space of directions and scale-multiplicative semigroups for restricted Burger-Mozes groups. We relate these general notions to intrinsic properties of the group. Among other things, we give a formula for the scale function; relate the space of directions to both the action on the tree and an action on a $\operatorname{CAT}(0)$ cube complex; and construct maximal scale-multiplicative semigroups from the space of directions.