  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 $\lim_{n\to\infty} f(n)/g(n)=1$.
