  We show that the topological complexity of an aspherical space $X$ is bounded
below by the cohomological dimension of the direct product $A\times B$,
whenever $A$ and $B$ are subgroups of $\pi_1(X)$ whose conjugates intersect
trivially. For instance, this assumption is satisfied whenever $A$ and $B$ are
complementary subgroups of $\pi_1(X)$. This gives computable lower bounds for
the topological complexity of many groups of interest (including semidirect
products, pure braid groups, certain link groups, and Higman's acyclic
four-generator group), which in some cases improve upon the standard lower
bounds in terms of zero-divisors cup-length. Our results illustrate an intimate
relationship between the topological complexity of an aspherical space and the
subgroup structure of its fundamental group.
