The topological complexity sequence of any group with infinite cohomological dimension is weakly increasing and unbounded, with growth estimates and exact asymptotics determined for finite groups of even order.
Minowa, On the topological complexity of non-simply connected spaces
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Topological complexity sequences of groups
The topological complexity sequence of any group with infinite cohomological dimension is weakly increasing and unbounded, with growth estimates and exact asymptotics determined for finite groups of even order.