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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 $$N$$.

## Sources

1. http://www.sciencedirect.com/science/article/pii/S0040938307000067

## See also

Large numbers in computers