Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914160425566208 |
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| author | Wang, Ke Zhang, Qiang Zhao, Dongxiao |
| author_facet | Wang, Ke Zhang, Qiang Zhao, Dongxiao |
| contents | In this paper, we first study the endomorphisms of free-abelian times surface groups and give a characterization of when they are injective and surjective. Then, we see that free-abelian times hyperbolic groups are Hopfian but not co-Hopfian. Moreover, we give a complete classification of fixed subgroups of endomorphisms in free-abelian times surface groups, which extends that of automorphisms. Finally, we study the endomorphisms of free-abelian times non-elementary torsion-free hyperbolic groups and give an equivalent condition for them to contain, up to isomorphism, finitely many end-fixed subgroups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups Wang, Ke Zhang, Qiang Zhao, Dongxiao Group Theory Geometric Topology In this paper, we first study the endomorphisms of free-abelian times surface groups and give a characterization of when they are injective and surjective. Then, we see that free-abelian times hyperbolic groups are Hopfian but not co-Hopfian. Moreover, we give a complete classification of fixed subgroups of endomorphisms in free-abelian times surface groups, which extends that of automorphisms. Finally, we study the endomorphisms of free-abelian times non-elementary torsion-free hyperbolic groups and give an equivalent condition for them to contain, up to isomorphism, finitely many end-fixed subgroups. |
| title | Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2511.12871 |