{"paper":{"title":"Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A dynamical lim-free criterion for hyperbolic group actions is equivalent to being mixed identity free.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ekaterina Rybak","submitted_at":"2026-05-13T22:24:20Z","abstract_excerpt":"We study the interplay between the algebraic and dynamical properties of groups that admit a general type action on a $\\delta$-hyperbolic space such that the induced action on the limit set of the Gromov boundary is faithful. We divide the class of such groups into two subclasses based on a dynamical criterion, which we call lim-free. We prove that satisfying the criterion is equivalent to a purely algebraic property of being mixed identity free, generalizing results from \\cite{FMMS} and \\cite{BM}. For the subclass of not lim-free groups, we give the rigidity result for all $3$-transitive fait"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"satisfying the criterion is equivalent to a purely algebraic property of being mixed identity free, generalizing results from FMMS and BM. For the subclass of not lim-free groups, we give the rigidity result for all 3-transitive faithful actions and bound the transitivity degree by 3.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The groups admit a general type action on a δ-hyperbolic space such that the induced action on the limit set of the Gromov boundary is faithful.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Lim-free dynamical criterion for faithful boundary actions on hyperbolic spaces is equivalent to the algebraic property of being mixed identity free, with transitivity degree bounded by 3 for non-lim-free groups.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A dynamical lim-free criterion for hyperbolic group actions is equivalent to being mixed identity free.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"154f8da298fada344e344e72101ed29a68c5d2932e3005e319d901e851882bdb"},"source":{"id":"2605.14159","kind":"arxiv","version":1},"verdict":{"id":"9b9c5641-05c1-4ebe-b9b8-b32bd090e921","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:57:48.963197Z","strongest_claim":"satisfying the criterion is equivalent to a purely algebraic property of being mixed identity free, generalizing results from FMMS and BM. For the subclass of not lim-free groups, we give the rigidity result for all 3-transitive faithful actions and bound the transitivity degree by 3.","one_line_summary":"Lim-free dynamical criterion for faithful boundary actions on hyperbolic spaces is equivalent to the algebraic property of being mixed identity free, with transitivity degree bounded by 3 for non-lim-free groups.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The groups admit a general type action on a δ-hyperbolic space such that the induced action on the limit set of the Gromov boundary is faithful.","pith_extraction_headline":"A dynamical lim-free criterion for hyperbolic group actions is equivalent to being mixed identity free."},"references":{"count":27,"sample":[{"doi":"","year":2023,"title":"[AMS] R. Aoun, P. Mathieu and C ¸ . Sert, Random walks on hyperbolic spaces: second order expansion of the rate function at the drift, J. ´Ec. polytech. Math.10(2023), 549–573. [AS] R. Aoun and C ¸ . ","work_id":"e67b02c8-9b40-4d52-ab26-4d6bab6d8551","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"Bass, Covering theory for graphs of groups,J","work_id":"8644bf0e-995d-4a6b-afdc-c88d78504370","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"[BEH] C. Bleak, L. Elliott, J. T. Hyde, Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated,J. Inst. Math. Jussieu23(2024), no. 6, 2825 –","work_id":"5383a2ed-8ada-4dbd-8519-89685ed210cb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"[BFFHZ] J. Belk, F. Fournier-Facio, J. Hyde, M. C. B. Zaremsky, Boone-Higman embed- dings ofAut(F n) and mapping class groups of punctured surfaces, arXiv:2503.21882v2. [BFGS] S.Balasubramanya, F. Fou","work_id":"b2fb7b5b-b9c1-4262-b6fc-a0e79e6793e1","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"[BM] A. Le Boudec, N. Matte Bon, Triple transitivity and non-free actions in dimension one,J. Lond. Math. Soc.105(2022), 2, 884 –","work_id":"03435cbb-5f0c-4cbd-b08e-3887fda71818","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":27,"snapshot_sha256":"6da15f186653e1d7caf536e3c95b7ba4dd724370c9fd58793a8763e4eedf6ee2","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}