Finite approximation of free groups I: the $F$-inverse cover problem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912707663364096 |
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| author | Auinger, K. Bitterlich, J. Otto, M. |
| author_facet | Auinger, K. Bitterlich, J. Otto, M. |
| contents | For a finite connected graph $\mathcal{E}$ with set of edges $E$, a finite $E$-generated group $G$ is constructed such that the set of relations $p=1$ satisfied by $G$ (with $p$ a word over $E\cup E^{-1}$) is closed under deletion of generators (i.e.~edges). As a consequence, every element $g\in G$ admits a unique minimal set $\mathrm{C}(g)$ of edges (the \emph{content} of $g$) needed to represent $g$ as a word over $\mathrm{C}(g)\cup\mathrm{C}(g)^{-1}$. The crucial property of the group $G$ is that connectivity in the graph $\mathcal{E}$ is encoded in $G$ in the following sense: if a word $p$ forms a path $u\longrightarrow v$ in $\mathcal{E}$ then there exists a $G$-equivalent word $q$ which also forms a path $u\longrightarrow v$ and uses only edges from their content; in particular, the content of the corresponding group element $[p]_G=[q]_G$ spans a connected subgraph of $\mathcal{E}$ containing the vertices $u$ and $v$. As the free group generated by $E$ obviously has these properties, the construction provides another instance of how certain features of free groups can be ``approximated'' or ``simulated'' in finite groups. As an application it is shown that every finite inverse monoid admits a finite $F$-inverse cover. This solves a long-standing problem of Henckell and Rhodes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_03273 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Finite approximation of free groups I: the $F$-inverse cover problem Auinger, K. Bitterlich, J. Otto, M. Group Theory Combinatorics 05C25, 05E18, 20M18, 20B99, 20F05, 20F65 For a finite connected graph $\mathcal{E}$ with set of edges $E$, a finite $E$-generated group $G$ is constructed such that the set of relations $p=1$ satisfied by $G$ (with $p$ a word over $E\cup E^{-1}$) is closed under deletion of generators (i.e.~edges). As a consequence, every element $g\in G$ admits a unique minimal set $\mathrm{C}(g)$ of edges (the \emph{content} of $g$) needed to represent $g$ as a word over $\mathrm{C}(g)\cup\mathrm{C}(g)^{-1}$. The crucial property of the group $G$ is that connectivity in the graph $\mathcal{E}$ is encoded in $G$ in the following sense: if a word $p$ forms a path $u\longrightarrow v$ in $\mathcal{E}$ then there exists a $G$-equivalent word $q$ which also forms a path $u\longrightarrow v$ and uses only edges from their content; in particular, the content of the corresponding group element $[p]_G=[q]_G$ spans a connected subgraph of $\mathcal{E}$ containing the vertices $u$ and $v$. As the free group generated by $E$ obviously has these properties, the construction provides another instance of how certain features of free groups can be ``approximated'' or ``simulated'' in finite groups. As an application it is shown that every finite inverse monoid admits a finite $F$-inverse cover. This solves a long-standing problem of Henckell and Rhodes. |
| title | Finite approximation of free groups I: the $F$-inverse cover problem |
| topic | Group Theory Combinatorics 05C25, 05E18, 20M18, 20B99, 20F05, 20F65 |
| url | https://arxiv.org/abs/2208.03273 |