{"paper":{"title":"$\\Phi$-Harmonic Functions on Discrete Groups and First $\\ell^\\Phi$-Cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Roman Panenko, Yaroslav Kopylov","submitted_at":"2013-11-10T07:09:44Z","abstract_excerpt":"We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\\ell^\\Phi(G)$, where $\\Phi$ is an $N$-function of class $\\Delta_2(0)\\cap \\nabla_2(0)$. In development of ideas of Puls and Martin--Valette, for a finitely generated group $G$, we introduce the discrete $\\Phi$-Laplacian and prove a theorem on the decomposition of the space of $\\Phi$-Dirichlet finite functions into the direct sum of the spaces of $\\Phi$-harmonic functions and $\\ell^\\Phi(G)$ (with an appropriate factorization). We also prove that if a finitely generated group $G$ has a finite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2246","kind":"arxiv","version":3},"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"}