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

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 $n$: $$I_{k,n}\sim \frac{(2k-1{)}^n}{nk!2^{k+1}}\mathrm{.}$$ Here $f(n)\sim g(n)$ means that ${\lim }_{n\to \mathrm{\infty }}f(n)\mathrm{/}g(n)=1$.