  In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting
the complexity of algorithms for autonomous robot motion. Here, $X$ stands for
the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the
minimal number of continuous rules which are needed to construct a motion
planning algorithm in $X$. We focus on the case when the space $X$ is
aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group
$\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$
can be characterised as the smallest integer $k$ such that the canonical
$\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to
E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the
$k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol
$E(\pi\times\pi)$ denotes the classifying space for free actions and
$E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions with
isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using
this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the
equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3,
{\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal
D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with
respect to the family of subgroups $\mathcal D$. We also introduce a Bredon
cohomology refinement of the canonical class and prove its universality.
Finally we show that for a large class of principal groups (which includes all
torsion free hyperbolic groups as well as all torsion free nilpotent groups)
the essential cohomology classes in the sense of Farber and Mescher are exactly
the classes having Bredon cohomology extensions with respect to the family
$\mathcal D$.
