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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.21936 |
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| _version_ | 1866912789808807936 |
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| author | Balasubramanya, Sahana Fernos, Talia |
| author_facet | Balasubramanya, Sahana Fernos, Talia |
| contents | We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $δ$-hyperbolic spaces with general type factors and associated subdirect products. This work is inspired by the classical theory of $S$-arithmetic lattices and the flourishing theory of acylindrically hyperbolic groups. In this paper - the first of three - we develop various fundamental results, explore elementary subgroups in higher rank, and exhibit a free vs abelian Tits Alternative. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-one setting.
The vastness of this class of groups is exhibited by recognizing that it contains $S$-arithmetic lattices with rank-one factors, acylindrically hyperbolic groups, colorable HHGs, groups with property (QT), and enjoys robust stability properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Acylindricity in Higher Rank, Part I : Fundamentals Balasubramanya, Sahana Fernos, Talia Group Theory 20F67, 20F65 We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $δ$-hyperbolic spaces with general type factors and associated subdirect products. This work is inspired by the classical theory of $S$-arithmetic lattices and the flourishing theory of acylindrically hyperbolic groups. In this paper - the first of three - we develop various fundamental results, explore elementary subgroups in higher rank, and exhibit a free vs abelian Tits Alternative. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-one setting. The vastness of this class of groups is exhibited by recognizing that it contains $S$-arithmetic lattices with rank-one factors, acylindrically hyperbolic groups, colorable HHGs, groups with property (QT), and enjoys robust stability properties. |
| title | Acylindricity in Higher Rank, Part I : Fundamentals |
| topic | Group Theory 20F67, 20F65 |
| url | https://arxiv.org/abs/2512.21936 |