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Ideals of equations for elements in a free group and Stallings folding

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  19 December 2024

Dario Ascari Open the ORCID record for Dario Ascari [Opens in a new window]
Dario Ascari*
Affiliation:
Department of Mathematics, University of the Basque Country, Leioa, Biscay, Spain
*
Email: ascari.maths@gmail.com
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Abstract

Let $H\le F$ be two finitely generated free groups. Given $g\in F$, we study the ideal $\mathfrak I_g$ of equations for g with coefficients in H, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in F. The ideal $\mathfrak I_g$ is a normal subgroup of $H*\langle x\rangle$, and it’s possible to algorithmically compute a finite normal generating set for $\mathfrak I_g$; we give a description of one such algorithm, based on Stallings folding operations. We provide an algorithm to find an equation in w(x)\in$\mathfrak I_g$ with minimum degree, i.e.  such that its cyclic reduction contains the minimum possible number of occurrences of x and x−1; this answers a question of A. Rosenmann and E. Ventura. More generally, we show how to algorithmically compute the set Dg of all integers d such that $\mathfrak I_g$ contains equations of degree d; we show that Dg coincides, up to a finite set, with either $\mathbb N$ or $2\mathbb N$. Finally, we provide examples to illustrate the techniques introduced in this paper. We discuss the case where ${\text{rank}}(H)=1$. We prove that both kinds of sets Dg can actually occur. We show that the equations of minimum possible degree aren’t in general enough to generate the whole ideal $\mathfrak I_g$ as a normal subgroup.

Keywords

geometric group theoryfree groupsequations over groupsStallings automataalgorithms in group theory

MSC classification

Primary: 20F70: Algebraic geometry over groups; equations over groups
Secondary: 20E05: Free nonabelian groups 20F65: Geometric group theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 42
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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