Polynomial time conjugacy in wreath products and free solvable groups

We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC0-many-one-reducible to the conjugacy problem in A ≀ B. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in A ≀ B to the power problems of A and B. Together with our first result, this yields a uniform TC0 solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also in free solvable groups.

We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable groups. For a finitely generated group we study the so-called power word problem (does a given expression u₁^{k₁} … u_d^{k_d}, where u₁, …, u_d are words over the group generators and k₁, …, k_d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u₁^{x₁} … u_d^{x_d} = v, where u₁, …, u_d,v are words over the group generators and x₁,…,x_d are variables, have a solution in the natural numbers). We prove that the power word problem for wreath products of the form G ≀ ℤ with G nilpotent and iterated wreath products of free abelian groups belongs to TC⁰. As an application of the latter, the power word problem for free solvable groups is in TC⁰. On the other hand we show that for wreath products G ≀ ℤ, where G is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is coNP-hard. For the knapsack problem we show NP-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product G ≀ ℤ, where G is uniformly efficiently non-solvable, is Σ₂^p-hard.

In this paper, we describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.

We show that the conjugacy problem in a wreath product \(A \wr B\) is uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. Moreover, if B is torsion free, the power problem for B can be replaced by the slightly weaker cyclic submonoid membership problem for B, which itself turns out to be uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in \(A \wr B\) if A is non-abelian.

We examine the relationship between the complexity of the word problem for a presentation and the complexity of the problem of determining the length of a shortest word equivalent to a given word. Our main result is that the length of the element represented by a word in a free solvable group can be determined in polynomial time.

We examine the relationship between the complexity of the word problem for a presentation and the complexity of the problem of determining the length of a shortest word equivalent to a given word. Our main result is that the length of the element represented by a word in a free solvable group can be determined in polynomial time.

We construct easy embeddings of relatively free groups (say the free Burnside group, the free solvable group) into finitely presented groups. We introduce a concept of verbal isoperimetric function of a group variety. We prove that if the verbal Dehn function of a relatively free group is bounded by a polynomial then the group can be embedded quasi-isometrically into a finitely presented group with polynomial isoperimetric function. We also construct an easy embedding of any Baumslag-Solitar solvable group into a finitely presented group with polynomial Dehn function.

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group G, knapsack (as well as the related subset sum problem) for the wreath product G \wr Z is NP-complete.

It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete [A. Myasnikov, V. Roman'kov, A. Ushakov and A. Vershik, The word and geodesic problems in free solvable groups, Trans. Amer. Math. Soc.362(9) (2010) 4655–4682] (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem. We show that the Geodesic Problem in the restricted wreath product of a finitely generated non-trivial group with a finitely generated abelian group containing ℤ2 is NP-hard and there exists a Polynomial Time Approximation Scheme for this problem. We also show that the Geodesic Problem in the restricted wreath product of two finitely generated non-trivial abelian groups is NP-hard if and only if the second abelian group contains ℤ2.

We study the computational complexity of the Word Problem (WP) in free solvable groups S r , d S_{r,d} , where r ≥ 2 r \geq 2 is the rank and d ≥ 2 d \geq 2 is the solvability class of the group. It is known that the Magnus embedding of S r , d S_{r,d} into matrices provides a polynomial time decision algorithm for WP in a fixed group S r , d S_{r,d} . Unfortunately, the degree of the polynomial grows together with d d , so the uniform algorithm is not polynomial in d d . In this paper we show that WP has time complexity O ( r n log 2 ⁡ n ) O(r n \log _2 n) in S r , 2 S_{r,2} , and O ( n 3 r d ) O(n^3 r d) in S r , d S_{r,d} for d ≥ 3 d \geq 3 . However, it turns out, that a seemingly close problem of computing the geodesic length of elements in S r , 2 S_{r,2} is N P NP -complete. We prove also that one can compute Fox derivatives of elements from S r , d S_{r,d} in time O ( n 3 r d ) O(n^3 r d) ; in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.

On the complexity of intersection and conjugacy problems in free groups

Algorithmic theory of free solvable groups: Randomized computations

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs:In this work we examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product G=H*AK with [H:A]≥[K:A] ≥2, the Schreier graph with respect to H or K turns out to be non-amenable if and only if [H:A]≥ 3. Moreover, for an HNN extension of the form G = , we show that the Schreier graph of G with respect to the subgroup H is non-amenable if and only if A ≠H ≠ φ(A).As application of these characterizations we show that under certain conditions the conjugacy problem in fundamental groups of finite graphs of groups with free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield another proof that the set where the conjugacy problem of the Baumslag group G1,2 is decidable in polynomial time is also strongly generic.

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.