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Our main result is that, for any word w in the free group on d generators, there exists $\\epsilon > 0$ such that if G is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $w\\colon G^d \\to G$ have Hausdorff dimension at most $d-\\epsilon$.\n  We conclude that the profinite completion of a group G as above satisfies no probabilistic identity. It is therefore randomly free; namely, for any d > 0, the probability that d randomly chosen el"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1706.08226","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-06-26T04:27:01Z","cross_cats_sorted":[],"title_canon_sha256":"fde15359e5e9a7af7929deb71cf8144dec920bc6fdf5f26824ff621d5ae947d2","abstract_canon_sha256":"edb81a59aaf016c3db03151533a8e9f9ef95ef406c78b3b2279bb34f7d464ba9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:43.151628Z","signature_b64":"leWExJAUuFHbk30d3imU/E0Zg4DElYA7FG53frjMrIvcQETOWKzkPdBaxQ/iRqDzNN4QVCIVRCN9VpnGqB1FBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"844b47d604c0d6f01fb18052558f73896666929bc747c9d19220ae8991095619","last_reissued_at":"2026-05-18T00:41:43.151167Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:43.151167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Words, Hausdorff dimension and randomly free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aner Shalev, Michael Larsen","submitted_at":"2017-06-26T04:27:01Z","abstract_excerpt":"We study fibers of word maps in finite, profinite, and residually finite groups. 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