{"paper":{"title":"Automorphism-invariant refinements of weakly branch actions via overlap functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Armando Martino","submitted_at":"2026-07-29T09:05:55Z","abstract_excerpt":"Let a finitely generated group $G$ act weakly branch on a locally finite rooted tree $T$ with boundary $\\partial T$. The rooted tree structure is encoded by the \\textit{overlap function}, which is our name for the Gromov product on the boundary: \\[\n  c(\\xi,\\eta)=|\\xi\\wedge\\eta|. \\] We axiomatise this function and show that when it is `admissible', one can recover the rooted tree.\n  By the boundary rigidity theorem of Lavreniuk and Nekrashevych, $\\operatorname{Aut}(G)$ acts canonically on $\\partial T$. We therefore form the automorphism symmetrisation of the overlap function: \\[\n  \\widehat c(\\x"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26644","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.26644/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"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"}