ABSTRACT

We investigate two possible definitions of group complexity
for finitely presented groups. Those are algebraic complexity
defined as the minimal length over all presentations of a
given group $G$, and geometric complexity which is the minimal
length of a ``geometric presentation'' of $G$ arising from
special spines $P$ such that $\pi_1(P)\cong G$. We show that
the algebraic complexity of a finite cyclic group $Z_n$
depends logarithmically on its order, and that the similar
estimate for geometric complexity of $Z_n$ holds modulo
a known number-theoretical conjecture.