Whitehead problems for words in Z m ˆ F n

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1 Whitehead problems for words in Z m ˆ F n Jordi Delgado Dept. Mat. Apl. III Universitat Politècnica de Catalunya, Manresa, Barcelona jorge.delgado@upc.edu jdelgado.upc@gmail.com January 15, 2013 Abstract We solve the Whitehead problem for automorphisms, monomorphisms and endomorphisms in Z m ˆ F n after giving an explicit description of each of these families of transformations. We generically call Whitehead problems for a finitely presented group G the problems consisting in, given two objects (of the same certain suitable kind O) in G, and a family F of transformations, decide whether there exists an element in F sending one object to the other. Specifically we will write WhPpO, Fq to mean the Whitehead problem with objects in O and transformations in F, i.e. WhPpO, Fq Dϕ P F such that o 1 ϕ ÞÑ o2? po1,o 2 in Oq. It is customary to include as a part of the problem, the search of one of such transformations, in case that there exists. So will we. The kind of objects in G usually considered include elements (i.e. words in the generators), subgroups, and conjugacy classes, as well as tuples of these; while the typical families of transformations (which we will always think acting on the right) are those of automorphisms, monomorphisms, epimorphisms and endomorphisms of G; we denote them respectively by Aut G, Mon G, Epi G and End G. The author thanks the hospitality of the Centre de Recerca Matemàtica (CRM- Barcelona) along the research programme on Automorphisms of Free Groups during which this preprint was finished; and gratefully acknowledge the support of Universitat Politècnica de Catalunya through the PhD grant number and the MEC (Spain) through project number MTM

2 This whole family of problems arise from the seminal WhPpF n, Aut F n q, where F n denotes the free group on n generators. It was proposed and solved by Whitehead in [6] using a (now classical) technique called peak-reduction. In this note we will deal with Whitehead problems for words in finitely generated free-abelian times free groups (see [4] for full details). In sake of notational easiness we will hereafter usually abbreviate G Z m ˆ F n. Concretely we will solve WhPpG, Aut Gq, WhPpG, Mon Gq and WhPpG, End Gq. It is not surprising that the (already solved) corresponding problems for Z m and F n emerge when considering Whitehead problems for G. For the free-abelian groups the problems considered become those of the existence of solutions (of certain type) for integer matrix equations of the form a X b. This can be easily decided using linear algebra. Proposition 1. Let m ě 1, then (i) WhPpZ m, Aut Z m q is solvable. (ii) WhPpZ m, Mon Z m q is solvable. (iii) WhPpZ m, End Z m q is solvable. The same problems for the free group F n are much more complicated. As mentioned above, the case of automorphisms was already solved by Whitehead back in the 30 s of the last century. The case of endomorphisms can be solved by writing a system of equations over F n (with unknowns being the images of a given free basis for F n ), and then solving it by the powerful Makanin s algorithm. Finally, the case of monomorphisms was recently solved by Ciobanu and Houcine. Theorem 2. Let n ě 2, then (i) [Whitehead, [6]] WhPpF n, Aut F n q is solvable. (ii) [Ciobanu-Houcine, [1]] WhPpF n, Mon F n q is solvable. (iii) [Makanin, [5]] WhPpF n, End F n q is solvable. So, the auto, mono and endo Whitehead problems (for words) are solvable for both Z m and F n. For G Z m ˆ F n though, these problems turn out to be more than the mere juxtaposition of the corresponding problems for its factors. That is because the endomorphisms of G are more than pairs of endomorphisms of Z m and F n as well. It is not difficult to obtain a complete description of them imposing the preservation of the (commutativity) relations defining G. 2

3 Proposition 3. The endomorphisms of G Z m ˆ F n are given by Ψ φ,q,p : pa, uq ÞÑ paq ` up, uφ a q where u u ab P Z n, Q and P are integer matrices, and φ a : F n Ñ F n is either (i) an endomorphism φ: F n Ñ F n (independent from a), or (ii) a map u ÞÑ w αpa,uq where w is a non-proper power in F n zt1u and αpa, uq al J ` uh J P Z for certain l P Z m zt0u and h P Z n. We will refer to them as type (I) and type (II) endomorphisms of G respectively. Note that if n 0 then type (I) and type (II) endomorphisms do coincide. Otherwise, it turns out that type (II) endomorphisms are a sort of degenerated case corresponding to a free contribution from the abelian part while all the injective and exhaustive endomorphisms of G are of type (I). Indeed, viewing Q as the endomorphism of Z m given by right multiplying by Q, we have the following quite natural characterization (note that the matrix P plays absolutely no role in this matter). Proposition 4. Let Ψ be an endomorphism of G Z m ˆ F n, with n ě 2. Then, (i) Ψ is a monomorphism if and only if it is of type (I) with φ a monomorphism of F n and Q a monomorphism of Z m (i.e. det Q 0). (ii) Ψ is an epimorphism if and only if it is of type (I) with φ an epimorphism of F n and Q an epimorphism of Z m (i.e. det Q 1). The hopfianity of Z m and F n together with this last proposition provide immediately the following results. Corollary 5. Z m ˆ F n is hopfian and not cohopfian. Corollary 6. An endomorphism of G Z m ˆ F n (n ě 2) is an automorphism if and only if it is of type (I) with φ P Aut pf n q and Q P GL m pzq. Now we have the ingredients to prove the main result of this note. Theorem 7. Let G Z m ˆ F n with m ě 1 and n ě 2, then (i) WhPpG, Aut Gq is solvable. (ii) WhPpG, Mon Gq is solvable. (iii) WhPpG, End Gq is solvable. 3

4 Sketch of the proof. We are given two elements pa, uq, pb, vq P G, and have to decide whether there exists an automorphism (resp. monomorphism, endomorphism) of Z m ˆ F n sending one to the other; and in affirmative case, find one of them. Using the previous descriptions for each type of transformations in Z m ˆ F n and separating the free-abelian and free parts, our problems reduce to deciding whether there exist integer matrices P, Q and a transformation φ of F n (Q and φ of certain kind depending on the case, see proposition 4) such that the two following independent conditions hold. uφ v aq ` up b Note that the subproblem associated to condition (1) becomes respectively the already solved WhPpF n, Aut F n q, WhPpF n, Mon F n q and WhPpF n, End F n q in the cases of autos, monos, and endos of type (I), and is straightforward to check for endos of type (II). Thus, if there is not any φ solving each of these problems (for F n ), then the corresponding problem (for G) has no solution either, and we are done. Otherwise, the decision method provides such a φ and our target reduces to solving the subproblem associated to condition (2): given arbitrary elements a P Z m and u P Z n, decide whether there exist integer matrices P and Q (satisfying det Q 0 in the case of monos, and det Q 1 in the case of autos) such that aq ` up b. If a 0 or u 0, these are well known results in linear algebra, otherwise write α gcdpaq 0 and µ gcdpuq 0. Then, the problems reduce to test whether the following linear system of equations, α x 1 ` µ y 1 b 1 /... (3) /- α x m ` µ y m b m has integral solutions x 1,..., x m, y 1,..., y m P Z (with no extra condition in the case of endos, satisfying px 1,..., x m q 0 in the case of monos, and satisfying gcdpx 1,..., x m q 1 in the case of autos). So, for the case of endos the decision is a standard argument in linear algebra. In the case of monomorphisms, the condition px 1,..., x m q 0 turns out to be superfluous and the same standard argument works, while the more involved case of autos become an exercise in elementary arithmetic and is decidable as well. Finally, observe that in any of the affirmative cases, we can use the description in proposition 3 to reconstruct a transformation Ψ (of the corresponding type) such that pa, uqψ pb, vq. *. (1) (2) 4

5 We note that, very recently, a new version of the classical peak-reduction theorem has been developed by M. Day [3] for an arbitrary partially commutative group (see also [2]). These techniques allow the author to solve the Whitehead problem for this kind of groups, in its variant relative to tuples of conjugacy classes and automorphisms. As far as we know, WhPpG, Mon Gq and WhPpG, End Gq remain unsolved for a general partially commutative group G. Our theorem 7 is a small contribution into this direction, solving these problems for free-abelian times free groups in a direct and selfcontained form. References [1] Ciobanu, L., and Houcine, A. The monomorphism problem in free groups. Archiv der Mathematik 94, 5 (2010), [2] Day, M. B. Peak reduction and finite presentations for automorphism groups of right-angled artin groups. Geometry & Topology 13 (Jan. 2009), [3] Day, M. B. Full-featured peak reduction in right-angled artin groups. arxiv: (Oct. 2012). [4] Delgado, J., and Ventura, E. Algorithmic problems for free-abelian times free groups. arxiv: (Jan. 2013). [5] Makanin, G. Equations in free groups (russian). Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), [6] Whitehead, J. H. C. On equivalent sets of elements in a free group. The Annals of Mathematics 37, 4 (Oct. 1936), ArticleType: research-article / Full publication date: Oct., 1936 / Copyright c 1936 Annals of Mathematics. 5