Complexity growth for topological black holes by holographic method

We consider the growth of the action for black hole spacetime with a fundamental string. Our interest is to find the difference of the behavior between black holes with three different topologies in the scenario of complexity-action conjecture. These black holes have positive, negative and zero curvatures. We would like to calculate the action growth of these systems with a probe fundamental string according to the complexity-action conjecture. We find that for the case where the black holes have the toroidal horizon structure this probe string behaves very differently from the other two cases.