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

The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that $$zx z^{-1} =y$$zxz-1=y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag---Solitar group $$\mathbf{{BS}}_{1,2}$$BS1,2 and the Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 (also known as Baumslag---Gersten group). The conjugacy problem in $${\mathbf{{BS}}}_{1,2}$$BS1,2 is complete for the circuit class $$\mathsf {TC}^0$$TC0. To the best of our knowledge $${\mathbf{{BS}}}_{1,2}$$BS1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 is an HNN-extension of $${\mathbf{{BS}}}_{1,2}$$BS1,2. Hence, decidability of the conjugacy problem in $$\mathbf{{G}}_{1,2}$$G1,2 outside the so-called black hole follows from Borovik et al. (Int J Algebra Comput 17(5/6):963---997, 2007). Decidability everywhere is due to Beese. Moreover, he showed exponential time for the set of elements which cannot be conjugated into $$\mathbf{{BS}}_{1,2}$$BS1,2 (Beese 2012). Here we improve Beese's result in two directions by showing that the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs, conjugacy in $${\mathbf{{G}}}_{1,2}$$G1,2 can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 by reducing the divisibility problem in power circuits to the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2. The complexity of the divisibility problem in power circuits is an open and interesting problem in integer arithmetic. We conjecture that it cannot be solved in elementary time because we can show that it cannot be solved in elementary time by calculating modulo values in power circuits.

The three fundamental decision problems posed by Max Dehn in 1912 are the word problem, the conjugacy problem and the isomorphism problem. Let G be a group and let u and v be elements of G. The word problem asks for an algorithm for deciding whether u = v. The conjugacy problem asks about the existence of an element g ∈ G which conjugates u to v, i.e., v = g −1 ug. A solution of the conjugacy problem clearly contains a solution of the word problem. The isomorphism problem asks whether two group presentations define isomorphic groups. The word and conjugacy problems received much attention in the literature. For a summary of results concerning the conjugacy problem until 1987 see [13] and references therein.

Conjugacy problem in groups of oriented geometrizable 3-manifolds

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.

There is an important class of finitely presented groups known as automatic groups, in which many computations can be undertaken efficiently. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. The normal form can also be used to enumerate group elements, and to compute the (rational) growth function of the group. The theory of automatic groups is developed [2], and the algorithms are described in more detail in [4]. Implementations of the algorithms are available in a software package of the author ([3]).The word-hyperbolic groups form an important subclass of the automatic groups. They are defined by the property that geodesic triangles in the Cayley graph of the group are uniformly thin. In other words, there is a constant e, such a that any point on one of the sides of such a triangle is within distance e of the union of the other two sides of the triangle. Small cancellation groups and groups acting discretely and cocompactly on hyperbolic space are examples of word-hyperbolic groups. Their theory is developed in [1].The word problem in a word-hyperbolic group can be solved by a Dehn algorithm in linear time. Furthermore, Epstein has shown that the conjugacy problem can be solved in time O(n ln n). However, both of these algorithms require a prior knowledge of the constant e, and it does not seem to be easy to compute this constant in general.In practice, the quadratic time algorithm for the word problem coming from the automatic groups procedures performs very effectively, and it would be useful to have alternatives for the conjugacy problem.The automatic groups machinery does provide such an algorithm (which works in the the intermediate class of biautomatic groups), but that unfortunately is exponential.Alternative algorithms for conjugacy testing of elements and subgroups are currently being developed and implemented by a research student of mine, Joe Marshall. They have the minor disadvanatage that their basic versions work only in torsion-free groups, but adaptations to the more general case seem possible. They include also a test for malnormality of a subgroup H of G, which means that g-1Hg∩H = 1 for all g ∈ G\H. This test is important in topological applications.They make essential use of the boundary of the group. The points on the boundary can be defined as the equivalence classes of infinite geodesic rays starting from the base point of the Cayley graph, where two such rays are equivalent if they remain a bounded distance apart. The precise bound involved here turns out to be another universal constant d of the Cayley graph, but unlike e it can be computed in practice relatively easily. Roughly speaking, a certain linear function of d provides us with a bound on the length of the elements g that we need to consider as possible conjugating elements when testing for conjugacy or malnormality. To make the algorithm practical, we need to restrict this set of potential conjugators considerably more than this, and Marshall has come up with various tricks for doing this.

<p>In this paper we investigate the structure of ring diagrams with periodic labels in groups with small cancellation conditions C(3)-T(6). These diagrams are used to solve the power conjugacy search problem in a cyclic subgroup and the problem of power conjugation. In groups of this class the first problem is positively solved. The second is formulated as follows.</p><p>To ascertain whether there are m, n, integers for which the degrees of v, w words with m, n exponents, respectively, are conjugated in the group G = (X; R). To solve this problem in the group with small cancellation conditions C(3)-T(6) it is sufficient to obtain upper bounds for the lengths of the boundary labels of the ring diagram. Therein lies this work.</p><p>Previously, it was proved that for any w word we can take a so-called normal form | word with the following property: each degree of normal form is R-irreducible. Studying of the ring diagrams with irreducible periodic labels manages to break a set of these diagrams into three classes. Working with one of these classes and using the periodicity of the boundary labels of diagram, it is possible to prove the periodicity of the layers of this diagram, and further limit the length of the boundary labels.</p><p>Thus, it turns out that to solve the power conjugacy problem is enough to use the finite known in advance number of checks for the conjugation of the certain degrees of v, w words. In the considered class of groups this problem is solved long ago.</p>

In 1911 M. Dehn posed the word and conjugacy problems for groups in general and provided algorithms which solved these problems for the fundamental groups of closed orientable two-dimensional manifolds. A crucial feature of these groups is that (with trivial exceptions) they are defined by a single relator r with the property that if s is a cyclic conjugate of r or r-1, with s ≠ r-1, there is very little cancellation in forming the product rs. Dehn’s algorithms have been extended to large classes of groups possessing presentations in which the defining relations have a similar small cancellation property. At first, investigations were concerned with the solution of the word problem for groups G presented as small cancellation quotients of a free group F. The theory was subsequently extended to the case where F is a free product, a free product with amalgamation, or an HNN extension. Moreover, strong results were obtained about algebraic properties; for example, one can classify torsion elements and commuting elements in small cancellation quotients.

We discuss a number of problems in relatively hyperbolic groups. We show that the word problem and the conjugacy (search) problem are solvable in linear and quadratic time, respectively, for a relatively hyperbolic group, whenever the corresponding problem is solvable in linear and quadratic time in each parabolic subgroup. We also consider the class R of finitely generated toral relatively hyperbolic groups. We show that groups from R are commutative transitive and generalize a theorem proved by Baumslag in [Bau67] to R. Moreover, we discuss two definitions of (fully) residually-C groups, the classical Definition 1.2.1 and a modified Definition 1.2.5. Building upon results obtained by Ol'shanskii [Ol'93] and Osin [Osi10], we prove the equivalence of the two definitions for C = R. This is a generalization of the similar result obtained by Ol'shanskii for the class of torsion-free hyperbolic groups. Let R be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [KM12] that a finitely generated fully residually- group G embeds into an iterated extension of centralizers of . We deduce from their theorem that every finitely generated fully residually- group embeds into a group from R. On the other hand, we give an example of a finitely generated torsion-free fully residually-H group that does not embed into a group from R; H is the class of hyperbolic groups.

The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups.

Decision problems and profinite completions of groups

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

A graph product is the fundamental group of a graph of groups Amongst the simplest examples are HNN groups and free products with amalgamation.The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical.For graph products over infinite graphs these conditions are insufficient: a further condition ensures that graph products over infinite graphs of bounded path length have solvable conjugacy problem.These results generalise the known ones for HNN groups and free products with amalgamation.

Let N 1 (N 2) be the normal closure of a finite symmetrized set R 1 (R 2, respectively) in a finitely generated free group F = F(A). As is known, if R i satisfies condition C(6), then the conjugacy problem is decidable in F/N i . In the paper, it is proved that, if one adds to condition C(6) on the set R 1 ∪ R 2 the atoricity condition for the presentation 〈A | R 1, R 2〉, then the conjugacy problem is decidable in the group F/N 1 ∩ N 2 as well. In particular, for the decidability of the conjugacy problem in F/N 1 ∩ N 2, it is sufficient to assume that the set R 1 ∪ R 2 satisfies condition C(7).

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.