Computably totally disconnected locally compact groups

TL;DR

This paper explores the algorithmic aspects of totally disconnected, locally compact groups, establishing computable presentations, dualities, and analyzing properties like the scale function and Cayley-Abels graphs.

Contribution

It introduces multiple equivalent approaches to defining computable presentations of t.d.l.c. groups and demonstrates their applicability to well-known groups, advancing the understanding of their algorithmic structure.

Findings

Established an algorithmic Stone-type duality for t.d.l.c. groups.

Proved several natural groups have computable presentations.

Identified conditions for uniqueness of computable presentations and examples with noncomputable scale functions.

Abstract

We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentations of t.d.l.c.\ groups, and show their equivalence. In the process, we obtain an algorithmic Stone-type duality between t.d.l.c.~groups and certain countable ordered groupoids given by the compact open cosets. We exploit the flexibility given by these different approaches to show that several natural groups, such as $\Aut(T_d)$ and $\SL_n(\QQ_p)$, have computable presentations. We show that many construction leading from t.d.l.c.\ groups to new t.d.l.c.\ groups have algorithmic versions that stay within the class of computably presented t.d.l.c.\ groups. This leads to further examples, such as $\PGL_n(\QQ_p)$. We study whether objects associated with computably t.d.l.c.\ groups are computable: the modular function, the scale function, and Cayley-Abels graphs in the compactly generated case. We give a criterion when computable presentations of t.d.l.c.~groups are unique up to computable isomorphism, and apply it to $\QQ_p$ as an additive group, and the semidirect product $\ZZ\ltimes \QQ_p$. We give (joint with Willis) an example of a computably t.d.l.c. group with noncomputable scale function.