# Coarse fundamental groups and box spaces

## TL;DR

This paper demonstrates that box spaces of finitely presented groups encode the normal subgroups used in their construction, enabling classification of groups via their box spaces and exploring various applications and examples.

## Contribution

It introduces a coarse fundamental group approach to distinguish box spaces, showing they detect normal subgroups and classify groups up to coarse equivalence, with new examples and applications.

## Key findings

- Box spaces detect normal subgroups up to isomorphism.
- Two finitely presented groups have coarsely equivalent box spaces iff they are commensurable via normal subgroups.
- Constructed examples of filtrations with bounded index differences but non-coarse equivalence.

## Abstract

We use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations $(N_i)$ and $(M_i)$ of a free group $F$ such that $M_i>N_i$ for all $i$ with $[M_i:N_i]$ uniformly bounded, but with $\Box_{(N_i)}F$ not coarsely equivalent to $\Box_{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first $\ell^2$ Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.