{"paper":{"title":"Self-similar groups and the zig-zag and replacement products of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ievgen Bondarenko","submitted_at":"2014-08-29T19:47:48Z","abstract_excerpt":"Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $\\Gamma_n$. We construct self-similar groups, whose graphs $\\Gamma_n$ can be represented as an iterated zig-zag product and graph powering: $\\Gamma_{n+1}=\\Gamma_n^k\\mathop{\\mbox{\\textcircled{$z$}}}\\Gamma$ ($k\\geq 1$). Also we construct self-similar groups, whose graphs $\\Gamma_n$ can be represented as an iterated replacement product and graph powering: $\\Gamma_{n+1}=\\Gamma_n^k\\mathop{\\mbox{\\textcircled{$r$}}}\\Gamma$ ($k\\geq 1$). This gives simple explicit examples of self-similar gr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.7115","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":""},"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"}