Quotient tests and random walks in computational group theory

For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few simple case studies of quotient tests for the classical identity, word, conjugacy problems in groups. We attempt to combine a rigorous analytic study with the assessment of algorithms from the practical point of view. It appears that, in case of finite quotient groups G/K, the efficiency of the quotient test very much depends on the mixing times for random walks on the Cayley graph of G/K. 1 Decision problems and quotient tests Let F = F (X) be a free group of rank m with basis X, R be a normal subgroup of F , and G = F/R. In this case we say that G is given by a presentation 〈X | R〉. By δ : F → G we denote the canonical epimorphism. Sometimes we will need to represent elements of G by (not necessary reduced) group words in X. To this end, denote by M = M(X) the set of all (not necessary reduced) words in the alphabet X±1 = X ∪ X−1, i.e., M is a free monoid with basis X±1. Denote by π :M → G the canonical epimorphism π : M → F and by ψ : M → G the canonical epimorphism ψ = π ◦ δ. Thus every word w ∈M (as well as w ∈ F ) represents an element ψ(w) ∈ G. A decision problem for a finitely generated group G given by a presentation G = 〈X | R〉 is a subset D ⊂M(X). A decision problem D ⊂M(X) is called decidable if there exists an algorithm to decide whether an arbitrary element of M(X) belongs to D or not. Instead of the problems over M(X) one can consider decision problems only over freely reduced words, that is, decision problems D ⊂ F (X). Since one can easily (in linear time) reduce a word inM(X) to its reduced form in F (X) these two decision problems are equivalent with respect to time complexity classes. In average-case (see [3]) or generic-case complexity (see [12]), where the measure