Delzant's T-invariant, Kolmogorov complexity and one-relator groups

Details Delzant's T-invariant, Kolmogorov complexity and one-relator groups Delzant's T-invariant, Kolmogorov complexity and one-relator groups

2003-05-25

arxiv.org/abs/math/0305353v3

Mathematics

Group Theory

A revised version, to appear in Comment. Math. Helv

Scientific paper

We prove that ``almost generically'' for a one-relator group Delzant's $T$-invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator groups and on the methods of the theory of Kolmogorov-Chaitin complexity. We also give a precise asymptotic estimate (when $k$ is fixed and $n$ goes to infinity) for the number $I_{k,n}$ of isomorphism classes of $k$-generator one-relator groups with a cyclically reduced defining relator of length $n$: \[ I_{k,n}\sim \frac{(2k-1)^n}{nk!2^{k+1}}. \] Here $f(n)\sim g(n)$ means that $\lim_{n\to\infty} f(n)/g(n)=1$.

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