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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.22841 |
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| _version_ | 1866909976993202176 |
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| author | Tsouvalas, Konstantinos |
| author_facet | Tsouvalas, Konstantinos |
| contents | We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear, hyperbolic group $Q$ that maps onto the free group of rank $2$ and $m\geq 2$, we construct a recursive sequence $(Γ_n)_{n\in \mathbb{N}}$ of torsion-free, hyperbolic $C'(\frac{1}{6})$ small cancellation groups, with the property that there is no algorithm determining the values $n\in \mathbb{N}$ such that $Γ_n$ has a quotient isomorphic to the direct product $Q^{m}$ of $m$-copies of $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Undecidability of epimorphisms onto products of hyperbolic groups Tsouvalas, Konstantinos Group Theory We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear, hyperbolic group $Q$ that maps onto the free group of rank $2$ and $m\geq 2$, we construct a recursive sequence $(Γ_n)_{n\in \mathbb{N}}$ of torsion-free, hyperbolic $C'(\frac{1}{6})$ small cancellation groups, with the property that there is no algorithm determining the values $n\in \mathbb{N}$ such that $Γ_n$ has a quotient isomorphic to the direct product $Q^{m}$ of $m$-copies of $Q$. |
| title | Undecidability of epimorphisms onto products of hyperbolic groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2512.22841 |