# Lower bounds on cubical dimension of $C'(1/6)$ groups

## TL;DR

This paper constructs specific finitely presented small cancellation groups that cannot act properly on any finite-dimensional CAT(0) cube complex, highlighting limitations in their geometric actions.

## Contribution

It introduces explicit examples of $C'(1/6)$ groups with bounded cubical dimension, advancing understanding of their geometric and combinatorial properties.

## Key findings

- Constructed finitely presented $C'(1/6)$ groups with no proper action on any $n$-dimensional CAT(0) cube complex.
- Demonstrated limitations of small cancellation groups in acting on low-dimensional cube complexes.
- Provided new insights into the geometric group theory of small cancellation groups.

## Abstract

For each $n$ we construct examples of finitely presented $C'(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.