Stallings foldings for rational subsets of automatic groups

LucÃ­a Asencio-MartÃ­n; John R. Britnell; Andrew Duncan; Dominik Francoeur; Sarah Rees

group theory

Stallings foldings for rational subsets of automatic groups

arXiv:2607.26284

summary

The paper presents an algorithm to build automata that recognize elements of a given submonoid or rational subset in an automatic group, using a convexity condition called L‑proximity, and applies the method to surface groups via small cancellation techniques.

Abstract

Let $G$ be an automatic group with associated regular language $L$ . We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset $K$ of $G$ . This builds on work of Kharlampovich, Miasnikov and Weil, on the case where $K$ is a subgroup of $G$ . Our construction succeeds, after sufficiently many iterations, whenever $K$ satisfies a certain convexity property, which we call $L$ -proximity. We show how to test whether the construction is complete in the case that $K$ is a submonoid; we have no such test for the general case of a rational subset $K$ . We focus particularly on the case of a surface group $G$ of genus $g > 1$ , where $L$ is the language of geodesic words in the standard generators. We use small cancellation theory to obtain a method for constructing $L$ -recognisable submonoids of $G$ .

24 pages, 6 figures

Topics & keywords

#automatic groups #rational subsets #submonoids #stallings foldings #surface groups #small cancellationautomatic groupregular languageStallings foldingL-proximityrational subsetsubmonoidgeodesic languagesmall cancellation