44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's commentsInternational audienceWe establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri typ
If 1 ; : : : ; n are limit groups and S 1 n is of type FP n /ޑ. then S contains a subgroup of finite index that is itself a direct product of at most n limit groups. This answers a question of Sela.
Let G be a group, and let r = r{t) be an element of the free product G * <£> of G with the infinite cyclic group generated by t. We say that the equation r(i) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * <£> to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to (the quotient of G * by the normal closure of r) being injective.In general it is not possible to find a solution to an arbitrary equation r(t) = 1 over an arbitrary group G. It is necessary to place some sort of restriction on the group, on the equation, or possibly on both. One possible restriction on the equation is that the exponent sum of t in r be non-zero. Under this hypothesis, but with no restriction on the group G, it is an open problem whether a solution over G always exists.It is known that a solution exists if G is either locally residually finite [6] or locally indicable [1,3,9], and other known results give solutions under restrictions on r. Levin [4] showed that a solution exists if t occurs in r only with positive exponent. Thus the simplest remaining case (up to conjugacy and inversion) is when r(t) has the form atbtct' 1 (a,b,ceG). Lyndon [5, Corollary 5.3] has solved this case under certain restrictions on the "coefficients" a, b, c. These restrictions are based on small cancellation theory, and concern the relations which can hold in G between a, b and c.It is the purpose of this note to remove the restrictions from Lyndon's result, and show that any equation of the form over any group G has a solution over G. Combined with Levin's theorem, this solves the problem whenever t occurs at most 3 times in r(t).Note that the above equation can be transformed to one of the form by applying the automorphism g>->g feeG), tr-ttb' 1 of G*(ty, so we are reduced to the case b=\. Also, if the equation has a solution over the subgroup G o of G generated by *Research supported by a William Gordon Seggie Brown Fellowship.
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SynopsisThe subsemigroup Singn of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r), the Stirling number of the second kind.
It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutationsgenerate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.
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