# Models for classifying spaces for $\mathbb{Z}\rtimes \mathbb{Z}$

## TL;DR

This paper constructs two specific models for the classifying space of infinite cyclic subgroups of the Klein bottle's fundamental group, providing examples outside the scope of existing general methods, especially for hyperbolic groups.

## Contribution

It introduces novel models for the classifying space of a particular group, expanding the understanding beyond previously known constructions for hyperbolic groups.

## Key findings

- Two explicit models for the classifying space are constructed.
- The models are applicable to the fundamental group of the Klein bottle.
- These models differ from existing general constructions.

## Abstract

We construct two models for the classifying space for the family of infinite cyclic subgroups of the fundamental group of the Klein bottle. These examples do not fit in general constructions previously done, for example, for hyperbolic groups.