Betti numbers \(b_k(G)\) result in an uncomputable fast-growing function as proposed by Alexander Nabutovsky and Shmuel Weinberger.[1]

Let \(H_k(G)\) be the \(k\)-th homology group of a group \(G\). Given a finite presentation of a group \(G\), we define its length as the sum of the lengths of all its relators plus the number of generators. The \(k\)-th Betti number of a finitely presented group \(G\) is defined as \(b_k(G) = \text{rank }H_k(G)\) (using the torsion-free rank). Given a nonnegative integer \(N\), we define the Betti number \(b_k(N)\) as the maximal finite \(b_k(G)\) where \(G\) is a finitely presented group with length at most \(N\).

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