Higher topological complexity of hyperbolic groups

Abstract We prove for non-elementary torsion-free hyperbolic groups $$\Gamma $$ Γ and all $$r\ge 2$$ r ≥ 2 that the higher topological complexity $${\textsf {TC}}_r(\Gamma )$$ TC r ( Γ ) is equal to $$r\cdot \mathrm {cd}(\Gamma )$$ r · cd ( Γ ) . In particular, hyperbolic groups satisfy the rationality conjecture on the $${\textsf {TC}}$$ TC -generating function, giving an affirmative answer to a question of Farber and Oprea. More generally, we show that the same conclusions hold for certain toral relatively hyperbolic groups.