Betti numbers of finitely presented groups and very rapidly growing   functions
  Define the length of a finite presentation of a group $G$ as the sum of lengths of all relators plus the number of generators. How large can be the $k$th Betti number $b_k(G)=$ rank $H_k(G)$ providing that $G$ has length $\leq N$ and $b_k(G)$ is finite? We prove that for every $k\geq 3$ the maximum $b_k(N)$ of $k$th Betti numbers of all such groups is an extremely rapidly growing function of $N$. It grows faster that all functions previously encountered in Mathematics (outside of Logic) including non-computable functions (at least those that are known to us). More formally, $b_k$ grows as the third busy beaver function that measures the maximal productivity of Turing machines with $\leq N$ states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines. We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function. Also, we outline a construction of a finitely presented group all of whose homology groups are either ${\bf Z}$ or trivial such that its Betti numbers form a random binary sequence. 
Alexander Nabutovsky and Shmuel Weinberger