2017/02/04 by David Kazhdan, Tamar Ziegler, Kazhdan, David +1
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1702.01308
openalex publication_date 2017/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field, G be an abelian group and r∈ \mathbb N. Let L be an infinite dimensional k-vector space. For any m∈ Endk(L) we denote by r(m)∈ [0,∞ ] the rank of m. We define by R(G,r,k)∈ [0,∞] the minimal R such that for any map A:G → Endk(L) with r(A(g'+g'')-A(g')-A(g''))≤ r, g',g''∈ G there exists a homomorphism χ:G→ Endk(L) such that r(A(g)-χ(g))≤ R(G, r, k) for all g∈ G. We show the finiteness of R(G,r,k) for the case when k is a finite field, G=V is a k-vector space V of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of \it Approximate Cohomology groups Hk\mathcal F (V,M) (which is a purely algebraic analogue of the notion of ε-representation (\citeep)) and interperate our result as a computation of the group H1\mathcal F (V,M) for some V-modules M.