Betti numbers are geometric invariants widely used in various branches of mathematics, including geometry, algebra, algebraic geometry, arithmetic geometry, and topological data analysis. It is originally defined for a topological space as the ranks of the Betti homology groups, i.e. the singular homology groups, but the construction also works for (discrete) groups $$G$$, thanks to the study of classification space $$BG$$ and the Eilenberg-MacLane space $$K(G,1)$$. The reader should be careful that the Betti homology of a discrete group is not defined as the Betti homology of the underlying topological space, which is trivial at positive degree.

## Applications in googology

Using betti numbers $$(b_k(G))_{k \in \mathbb{N}}$$ for finitely presented groups $$G$$, Alexander Nabutovsky and Shmuel Weinberger constructed a system $$(b_k(N))_{k \in \mathbb{N}}$$ of functions including uncomputable fast-growing functions.

Let $$H_k(G)$$ be the $$k$$-th homology group of a group $$G$$ in the sense explained above. Given a finite presentation of a group $$G$$ (with a fixed presentation), 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 number $$b_k(N)$$ as the maximum of finite Betti numbers $$b_k(G)$$ where $$G$$ runs theough finitely presented groups with length at most $$N$$.

## Sources

1. Betti Number in Wolfram Mth World.
2. Alexander Nabutovskyab and Shmuel Weinbergerc, Betti numbers of finitely presented groups and very rapidly growing functions, Topology, Volume 46, Issue 2, March 2007, pp 211--223.