Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph