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

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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 Iₖ,ₙ of isomorphism classes of k-generator one-relator groups with a cyclically reduced defining relator of length n: Iₖ,ₙ (2k-1)ⁿnk!2ᵏ⁺¹. Here f(n) g(n) means that ₙ→∞ f(n)/g(n)=1.