# Coarse structures on groups defined by $T$-sequences

## TL;DR

This paper investigates the coarse geometric properties of groups equipped with topologies induced by $T$-sequences, showing infinite asymptotic dimension and conditions for having one end, using Hamming space embeddings.

## Contribution

It introduces a new coarse structure on groups based on $T$-sequences and analyzes their asymptotic dimension and end structure, revealing infinite asymptotic dimension and conditions for one end.

## Key findings

- Asymptotic dimension of the coarse group is infinite for all non-trivial $T$-sequences.
- The coarse group has exactly one end if the $T$-sequence generates the group.
- Embeddings of Hamming space are used to establish these properties.

## Abstract

A sequence $(a_{n}) $ in an Abelian group is called a $T$-sequence if there exists a Hausdorff group topology on $G$ in which $(a_{n}) $ converges to $0$. For a $T$-sequence $(a_{n}) $, $\tau_{(a_{n}) } $ denotes the strongest group topology on $G$ in which $(a_{n}) $ converges to $0$. The ideal $\mathcal{I}_{(a_{n})} $ of all precompact subsets of $(G, \tau_{(a_{n}) } )$ defines a coarse structure on $G$ with base of entourages $\{(x, y): x-y \in P \}$, $P\in\mathcal{I}_{(a_{n})}. $ We prove that $asdim \ \ (G, \mathcal{I}_{(a_{n}) }) =\infty $ for every non-trivial $T$-sequence $(a_{n})$ on $G$, and the coarse group $(G, \mathcal{I}_{(a_{n}) })$ has 1 end provided that $(a_{n}) $ generates $G$. The keypart play asymorphic copies of the Hamming space in $(G, \mathcal{I}_{(a_{n})})$.