Abstract
In 2007, Nabutovsky and Weinberger provided a solution to a long-standing problem: to find naturally defined functions that grow faster than any function with Turing degree of unsolvability 0'. They considered the functions bksuch that, for a natural integer N, bk(N) is the rank of the kth homology group Hk(G) of maximum finite rank, among the finitely presented groups G with presentation length ≤ N. They proved that, for k ≥ 3, function bkgrows as the third busy beaver function, and so grows faster than any function with degree of unsolvability 0″.Can more be said about these functions bk? We give some results on the function b2, we study the challenge of computing Hk(G) for a finitely presented group G, and we compute bk(N) for small values of N.
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