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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.18742 |
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| _version_ | 1866918305996996608 |
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| author | Alekseev, Vadim Bradford, Henry |
| author_facet | Alekseev, Vadim Bradford, Henry |
| contents | We introduce the notion of a ``sofic $\mathcal{C}$-action'' of one group on another by automorphisms, for $\mathcal{C}$ a class of groups. We show that if $\mathcal{C}$ is the class of (i) sofic, (ii) hyperlinear, (iii) linear sofic or (iv) weakly sofic groups, then the class $\mathcal{C}$ is closed under taking semidirect products with sofic $\mathcal{C}$-action. We use this to construct a wide variety of new examples of groups in the classes (i)-(iv), many of them arising as ``halo products'' in the sense of Genevois-Tessera. We have a parallel set of results producing new examples of semidirect products which are locally embeddable into finite groups. Our framework also unifies existing results in the literature, due to Hayes-Sale; Brude-Sasyk and Gao-Kunnawalkam Elayavalli-Patchell. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18742 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sofic actions, halo products, and metric approximations of groups Alekseev, Vadim Bradford, Henry Group Theory We introduce the notion of a ``sofic $\mathcal{C}$-action'' of one group on another by automorphisms, for $\mathcal{C}$ a class of groups. We show that if $\mathcal{C}$ is the class of (i) sofic, (ii) hyperlinear, (iii) linear sofic or (iv) weakly sofic groups, then the class $\mathcal{C}$ is closed under taking semidirect products with sofic $\mathcal{C}$-action. We use this to construct a wide variety of new examples of groups in the classes (i)-(iv), many of them arising as ``halo products'' in the sense of Genevois-Tessera. We have a parallel set of results producing new examples of semidirect products which are locally embeddable into finite groups. Our framework also unifies existing results in the literature, due to Hayes-Sale; Brude-Sasyk and Gao-Kunnawalkam Elayavalli-Patchell. |
| title | Sofic actions, halo products, and metric approximations of groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2601.18742 |