The scale function and lattices

TL;DR

This paper demonstrates that the properties of contraction subgroups and the scale function in a totally disconnected, locally compact group are determined by a lattice within it, linking lattice properties to the larger group's structure.

Contribution

It establishes that contraction subgroups and the scale function in G are fully determined by their restrictions to a lattice H, revealing how lattice properties influence the ambient group.

Findings

Contraction subgroups in G are determined by H

Scale function values in G are determined by H

Lattice properties imply group properties

Abstract

It is shown that, given a lattice H in a totally disconnected, locally compact group G, the contraction subgroups in G and the values of the scale function on G are determined by their restrictions to H. Group theoretic properties intrinsic to the lattice, such as being periodic or infinitely divisible, are then seen to imply corresponding properties of G.