Computability of finite quotients of finitely generated groups

We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem

We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

Operator algebras with hyperarithmetic theory

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

Computability of Følner sets

We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions — with subrecursive distortion functions — of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets. Moreover, we obtain some known and some new upper bounds for the Følner function for these particular extensions.

A combination theorem for affine tree-free groups

Let [Formula: see text] be an ordered abelian group. We show how a group admitting a free affine action without inversions on a [Formula: see text]-tree admits a natural graph of groups decomposition, where vertex groups inherit actions on [Formula: see text]-trees. We introduce a stronger condition (essential freeness) on an affine action and apply recent work of various authors to deduce that a finitely generated group admitting an essentially free affine action on a [Formula: see text]-tree is relatively hyperbolic with nilpotent parabolics, is locally relatively quasiconvex, and has solvable word, conjugacy and isomorphism problems. Conversely, given a graph of groups satisfying certain conditions, we show how an affine action of its fundamental group can be constructed. Specialising to the case of free affine actions, we obtain a large class of groups that have a free affine action on a [Formula: see text]-tree but that do not act freely by isometries on any [Formula: see text]-tree. We also give an example of a group that admits a free isometric action on a [Formula: see text]-tree but which is not residually nilpotent.

Iterated effective embeddings of abelian p-groups

This paper contributes to the theory of recursively presented (see Higman [Subgroups of finitely presented groups, Proc. R. Soc. Ser. A 262 (1961) 455–475]) infinitely generated abelian groups with solvable word problem. Mal'cev [On recursive Abelian groups, Dokl. Akad. Nauk SSSR 146 (1962) 1009–1012] and independently Rabin [Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960) 341–360] initiated the study of such groups in the early 1960's. In this paper, we develop a technique that we call iterated effective embeddings. The significance of our new technique is that it extends existing methods from the realm of iterated 0″ arguments to iterated 0‴ ones. This is a new phenomenon in computable algebra. We use this technique to confirm a 30 year-old conjecture of Ash, Knight and Oates [Recursive abelian p-groups of small length, https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ]. More specifically, Ash, Knight and Oates [Recursive abelian p-groups of small length. https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ] conjectured that there exists a computable reduced abelian p-group of Ulm type ω such that its effective invariants, defined using limitwise monotonic functions, cannot be found uniformly. We construct a computable reduced abelian p-group of Ulm type ω where its invariants are at the maximum possible level of non-uniformity. The result confirms the conjecture in a strong way, and it provides us with an explanation of why computable reduced p-groups of Ulm type ω seem hard to classify in general. We also use p-basic trees and their iterated embeddings to solve a problem posed in [W. Calvert, D. Cenzer, V. S. Harizanov and A. Morozov, Effective categoricity of abelian p-groups, Ann. Pure Appl. Logic 159(1–2) (2009) 187–197].

QUASI-ISOMETRY AND FINITE PRESENTATIONS OF LEFT CANCELLATIVE MONOIDS

We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.

A FINITELY PRESENTED GROUP WITH ALMOST SOLVABLE CONJUGACY PROBLEM

The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ > 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd (m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.