The well-known theorem of Eilenberg and Ganea [Ann. Math.�65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space as the cohomological dimension of the group . In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces . One of our main results states that in the case when the group is hyperbolic in the sense of Gromov, the topological complexity either equals or is by one larger than the cohomological dimension of . We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.