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This paper presents simulation and separation results on the computational complexity of cellular automata (CA) in the hyperbolic plane. It is shown that every t(n)-time nondeterministic hyperbolic CA can be simulated by an O(t3(n))-time deterministic hyperbolic CA. It is also shown that for any computable functions t1(n) and t2(n) such that limn驴驴 (t1(n))3/t2(n) = 0, t2(n)-time hyperbolic CA are strictly more powerful than t1(n)-time hyperbolic CA. This time hierarchy holds for both deterministic and nondeterministic cases. As for the space hierarchy, hyperbolic CA of space s(n) + 驴(n) are strictly more powerful than those of space s(n) if 驴(n) is a function not bounded by O(1).