# Explicit $\ell^1$-efficient cycles and amenable normal subgroups

## TL;DR

This paper provides an elementary construction of explicit $ ext{l}^1$-efficient cycles in group homology, demonstrating the isometric projection property for quotients by amenable normal subgroups, based on Gromov's theorems.

## Contribution

It offers a simplified, explicit method for constructing $ ext{l}^1$-efficient cycles, clarifying Gromov's implicit results on homology projections.

## Key findings

- Projection is isometric on homology with respect to $ ext{l}^1$-semi-norm.
- Provides an elementary explicit construction of efficient cycles.
- Clarifies the diffusion of cycles in the context of amenable normal subgroups.

## Abstract

By Gromov's mapping theorem for bounded cohomology, the projection of a group to the quotient by an amenable normal subgroup is isometric on group homology with respect to the $\ell^1$-semi-norm. Gromov's description of the diffusion of cycles also implicitly produces efficient cycles in this situation. We present an elementary version of this explicit construction.