Negative type functions on groups with polynomial growth

TL;DR

This paper demonstrates the existence of continuous, proper, negative definite functions with polynomial growth on groups with polynomial growth, closely matching the group's homogeneous dimension.

Contribution

It establishes the construction of negative definite functions with polynomial growth dimensions arbitrarily close to the group's homogeneous dimension on certain groups.

Findings

Existence of negative definite functions with specified polynomial growth

Construction applicable to groups with polynomial growth and homogeneous dimension

Negative definite functions can approximate the group's growth dimension

Abstract

The aim of this work is to show that on a locally compact, second countable, compactly generated group $G$ with polynomial growth and homogeneous dimension $d_h$, there exist a continuous, proper, negative definite function $\ell$ with polynomial growth dimension $d_\ell$ arbitrary close to $d_h$.