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It is proved that when $r<n,$ a group action of $\\mathrm{SL}_{n}(\\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivial if and only if the induced group homomorphism $\\mathrm{SL}_{n}(% \\mathbb{Z})\\rightarrow \\mathrm{Out}(\\pi _{1}(M))$ is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. Especially, when $\\pi _{1}(M)$ is nilpotent, the group $\\mathrm{SL}_{n}(% \\mathbb{Z})$ cannot act nontrivially on $M$ when $r<n.$ This confirms a conjec"},"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":"1609.07699","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-09-25T04:17:44Z","cross_cats_sorted":["math.DS","math.GT"],"title_canon_sha256":"6b385381440e876a3273b58e31b991abc04bb13d17e4473ae3a300a150710184","abstract_canon_sha256":"a92a6989fb551b08e4c916b1c9cce750fed25c566f4e7623feca09856ff17728"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:01.278981Z","signature_b64":"+eVKdkEUhrOFiYRcg9GADk4q2m4fEm/FTSJDXGNk8W9aVHX4tMl/8fzmmhx+dRFuBqTifVZGKnPEYvwZuWaHAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcd8bc4c172128c01d905711806c155cb0c5d5d19bbf9e42b85cf67bb9fccb90","last_reissued_at":"2026-05-18T00:07:01.278390Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:01.278390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The action of matrix groups on aspherical manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.AT","authors_text":"Shengkui Ye","submitted_at":"2016-09-25T04:17:44Z","abstract_excerpt":"Let $\\mathrm{SL}_{n}(\\mathbb{Z})$ $(n\\geq 3)$ be the special linear group and $M^{r}$ be a closed aspherical manifold. 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