Ideals of equations for elements in a free group and Stallings folding
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916807748616192 |
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| author | Ascari, Dario |
| author_facet | Ascari, Dario |
| contents | Let $F$ be a finitely generated free group and let $H\le F$ be a finitely generated subgroup. Given an element $g\in F$, 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 we provide an algorithm, based on Stallings folding operations, to compute a finite set of generators for $\mathfrak{I}_g$ as a normal subgroup.
We provide an algorithm to find an equation in $\mathfrak{I}_g$ with minimum degree, i.e. an equation $w(x)$ 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 provide an algorithm that, given $d\in\mathbb{N}$, determines whether $\mathfrak{I}_g$ contains equations of degree $d$ or not, and we give a characterization of the set of all the equations of that specific degree. We define the set $D_g$ of all integers $d$ such that $\mathfrak{I}_g$ contains equations of degree $d$; we show that $D_g$ coincides, up to a finite set, either with the set of non-negative even numbers or with the set of natural numbers.
Finally, we provide examples to illustrate the techniques introduces in this paper. We discuss the case where $\text{rank}(H)=1$. We prove that both kinds of sets $D_g$ can actually occur. The examples also 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_04759 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ideals of equations for elements in a free group and Stallings folding Ascari, Dario Group Theory 20F70, 20E05 (20F65) Let $F$ be a finitely generated free group and let $H\le F$ be a finitely generated subgroup. Given an element $g\in F$, 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 we provide an algorithm, based on Stallings folding operations, to compute a finite set of generators for $\mathfrak{I}_g$ as a normal subgroup. We provide an algorithm to find an equation in $\mathfrak{I}_g$ with minimum degree, i.e. an equation $w(x)$ 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 provide an algorithm that, given $d\in\mathbb{N}$, determines whether $\mathfrak{I}_g$ contains equations of degree $d$ or not, and we give a characterization of the set of all the equations of that specific degree. We define the set $D_g$ of all integers $d$ such that $\mathfrak{I}_g$ contains equations of degree $d$; we show that $D_g$ coincides, up to a finite set, either with the set of non-negative even numbers or with the set of natural numbers. Finally, we provide examples to illustrate the techniques introduces in this paper. We discuss the case where $\text{rank}(H)=1$. We prove that both kinds of sets $D_g$ can actually occur. The examples also 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. |
| title | Ideals of equations for elements in a free group and Stallings folding |
| topic | Group Theory 20F70, 20E05 (20F65) |
| url | https://arxiv.org/abs/2207.04759 |