Progress around the Boone–Higman conjecture

Finitely Determined Members of Varieties of Groups and Rings

Embedding Algebras with Solvable Word Problems in Simple Algebras - Some Boone-Higman Type Theorems

A finitely presented monoid which has solvable word problem but has no regular complete presentation

Embeddings into Finitely Generated Simple Groups Which Preserve the Word Problem

The group described in the title is obtained as a quotient of a center-by-metabelian group constructed by P. Hall. It is well known that a residually finite group which is finitely presented has a solvable word problem (V. H. Dyson [1], A. W. Mostowski [6]). It is also known (see the last sentence in V. H. Dyson [1]) although perhaps not well enough (see W. Magnus [5, p. 307]) that a residually finite group which is only finitely generated may have an unsolvable word problem. Herein we exhibit such a group which moreover is soluble, indeed it is center-by-metabelian. This is in some sense a minimal soluble example since all finitely generated metabelian groups have a solvable word problem. (This is an easy consequence of the fact that finitely generated free metabelian groups satisfy the maximal condition on normal subgroups (P. Hall [2]) and thus any finitely generated metabelian group can be almost finitely presented in the sense of Mostowski [6] and is residually finite (P. Hall [3]).) Credit must be given to Peter Neumann who pointed out the appropriate example to be modified. Our example uses a recursive function f with a nonrecursive range in a manner similar to the way Higman [5] exhibited a finitely presented group with an unsolvable word problem. Let G be the center-by-metabelian group constructed by P. Hall in ([2, p. 434]). In the notation of P. Hall, G is generated by a and b and defined by [bi, bj, bk] = 1 where bi = a-ibai, i, j, k = 0, 1, ... Ci; = Ci+kj+ where cij = [bj, bi], j > i, i, j, k = 0, ?1, The center C of G is free abelian with basis dl, d2, i i where dr = cii+, r = 1,2 ,i= 0, ?1, *. Received by the editors March 12, 1973. AMS (MOS) subject classifications (1970). Primary 20E25, 20F10; Secondary 20E1 5.

It has long been known that every free associative algebra can be embedded in a skew field [11]; in fact there are many different embeddings, all obtainable by specialization from the ‘universal field of fractions’ of the free algebra (cf. [5, Chapter 7]). This makes it reasonable to call the latter the free field; see §2 for precise definitions. The existence of this free field was first established by Amitsur [1], but his proof is rather indirect and does not provide anything like a normal form for the elements of the field. Actually one cannot expect to find such a normal form, since it does not even exist in the field of fractions of a commutative integral domain, but at least one can raise the word problem for free fields: Does there exist an algorithm for deciding whether a given expression for an element of the free field represents zero?Now some recent work has revealed a more direct way of constructing free fields ([4], [5], [6]), and it is the object of this note to show how this method can be used to solve the word problem for free fields over infinite ground fields. In this connexion it is of interest to note that A. Macintyre [9] has shown that the word problem for skew fields is recursively unsolvable. Of course, every finitely generated commutative field has a solvable word problem (see e.g. [12]).The construction of universal fields of fractions in terms of full matrices is briefly recalled in §2, and it is shown quite generally for a ringRwith a field of fractions inverting all full matrices, that if the set of full matrices overRis recursive, then the universal field has a solvable word problem. This holds more generally if the precise set of matrices overRinverted over the field is recursive, but it seems difficult to exploit this more general statement.

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].

We show that every countable group H with solvable word problem can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G.

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.

Exploiting a new technique for proving undecidability results developed by A. Sattler-Klein in her doctoral dissertation (1996) it is shown that it is undecidable in general whether or not a finitely presented monoid with a solvable word problem has finite derivation type (FDT). This improves upon the undecidability result of R. Cremanns and F. Otto (1996), which was based on the undecidability of the word problem for the finitely presented monoids considered.

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.

Novikov groups were introduced as examples of finitely presented groups with unsolvable conjugacy problem. It was Bokut who showed that each Novikov group has a standard basis and thus a solvable word problem. Further, he showed that for every recursively enumerable degree of unsolvability d there is a Novikov group whose conjugacy problem is of degree d. In the present work, we show that Novikov groups are also right-orderable, thus exhibiting the first known examples of finitely presented right-orderable groups with solvable word problem and unsolvable conjugacy problem.

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).

In this paper, we exhibit an example of a variety with only one binary operation having recursive base, undecidable equational theory and solvable word problem.

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 torsion and conjugacy theorems are proved for any group presented as a graph product. The graphs over which some graph product has a solvable word problem are characterised. Conditions are then given for the solvability of the word and order problems and also for the extended word problem for cyclic subgroups of any graph product. These results generalise the known ones for HNN groups and free products with amalgamation.