# Polynomially weighted $\ell^p$-completions and group homology

## TL;DR

This paper introduces polynomially weighted ^p-norms on group bar complexes and shows that for groups of polynomial or exponential growth, the resulting homology is independent of p in (1, ).

## Contribution

It defines new weighted ^p-norms on group homology complexes and proves p-independence of homology for certain groups.

## Key findings

- Homology of weighted complexes is p-independent for polynomial/exponential growth groups.
- Weighted ^p-norms are introduced on bar complexes.
- Homology results apply to finitely generated groups with specific growth conditions.

## Abstract

We introduce polynomially weighted $\ell^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,\infty)$.