The semi-simple theory of higher rank acylindricity
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911341670825984 |
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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. Inspired by the classical theory of ($S$-arithmetic) lattices and the flourishing theory of acylindrically hyperbolic groups, we show that, up to virtual isomorphism, finitely generated groups in this class enjoy a strongly canonical product decomposition. This semi-simple decomposition also descends to the outer-automorphism group, allowing us to give a partial resolution to a recent conjecture of Sela. We also develop various structure results including a free vs abelian Tits' Alternative, and connections to lattice envelopes. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-1 setting.
The vastness of this class of groups is exhibited by recognizing that it contains, for example, $S$-arithmetic lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, groups with property (QT), and is closed under direct products, passing to (totally general type) subgroups, and finite index over-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04838 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The semi-simple theory of higher rank acylindricity Balasubramanya, Sahana Fernos, Talia Group Theory General Topology 20F67, 20F65, 20F28 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. Inspired by the classical theory of ($S$-arithmetic) lattices and the flourishing theory of acylindrically hyperbolic groups, we show that, up to virtual isomorphism, finitely generated groups in this class enjoy a strongly canonical product decomposition. This semi-simple decomposition also descends to the outer-automorphism group, allowing us to give a partial resolution to a recent conjecture of Sela. We also develop various structure results including a free vs abelian Tits' Alternative, and connections to lattice envelopes. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-1 setting. The vastness of this class of groups is exhibited by recognizing that it contains, for example, $S$-arithmetic lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, groups with property (QT), and is closed under direct products, passing to (totally general type) subgroups, and finite index over-groups. |
| title | The semi-simple theory of higher rank acylindricity |
| topic | Group Theory General Topology 20F67, 20F65, 20F28 |
| url | https://arxiv.org/abs/2407.04838 |