In this paper we study topological invariants of a class of random groups.
Namely, we study right angled Artin groups associated to random graphs and
investigate their Betti numbers, cohomological dimension and topological
complexity. The latter is a numerical homotopy invariant reflecting complexity
of motion planning algorithms in robotics. We show that the topological
complexity of a random right angled Artin group assumes, with probability
tending to one, at most three values. We use a result of Cohen and Pruidze
which expresses the topological complexity of right angled Artin groups in
combinatorial terms. Our proof deals with the existence of bi-cliques in random
graphs.