Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

Abstract

We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips' classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in all hyperbolic groups (possibly with torsion). Additionally, we can deal with quasi-isometrically embeddable rational constraints. Introduction The equations problemGiven a group G, the equations problem in G consists in deciding algorithmically whether a system of equations with constants has a solution in G or not. An equation is an equality w = 1 where w is a word on a set of variables an their inverses together with constants taken from G. When inequations (i. e. negation of equations) are allowed, we call this problem the problem of equations and inequations. In other words, the problem of equations and inequations is equivalent to the decidability of the existential (or universal) theory of G with constants in G.A solution to the equations problem is quite powerful, as it vastly generalises the word problem, the conjugacy problem, the simultaneous conjugacy problem, etc. When G is abelian, the problem of equations and inequations is easily solved using linear algebra. But already, if G a free 3-step nilpotent group of rank 2, the equations problem is undecidable ([Tru95], see also [Rom79b]). This is based on Matiyasevich's Theorem saying that one cannot decide the solubility of polynomial equations in Z [Mat70]. For a free 2-step nilpotent group, the equations problem is solvable if and only if one can decide the solubility of polynomial equations over Q [Rom79b]. Additionally, if G is a non-commutative free metabelian group, the equations problem in G is unsolvable [Rom79a], but the problem of equations and inequations without constants is solvable [Cha95].The problem of equations and inequations in free groups is natural, and has attracted attention of many people (Lyndon, Appel, Lorents...). The solution of this problem by Makanin in 1982 [Mak82] (with the appropriate correction in [Mak84]) certainly constitutes a milestone in the theory. It has been a source of inspiration for Rips for his study of group actions on R-trees, and his solution to Morgan and Shalen's conjecture [BF95,GLP94], which found applications in many branches of geometry. It has been a decisive step towards algorithmic, and theoretical description of the set of homomorphisms of a group into a free group (Razborov [Raz84]). 1It has also been, together with these developments, a prelude to the far reaching recent solutions to Tarski's problems on the elementary theory of free groups, by Sela [Sel06], and by Kharlampovich and Miasnikov [KM06]. Makanin's algorithm is also the basis of Rips and Sela's solution to the equations problem in torsion free hyperbolic groups that they manage to reduce to the equations problem in a free group [RS95]. Finally, it is crucial in Sela's solution of the isomorphism problem for torsion free hyperbolic groups with f...

On suspensions and conjugacy of hyperbolic automorphisms

“…In [Br2] Brinkmann gave several examples with different behaviors. In particular, the solution to the isomorphism problem of hyperbolic groups will not reveal all isomorphisms between suspensions, and since the fibers are exponentially distorted in the suspensions, the usual rational tools (see [D,DGu1]) do not work for solving the isomorphism problem with such a preservation constraint. One can thus merely detect the existence of one isomorphism (say ι), but for investigating the existence of an isomorphism with the aforementioned properties, one is led to consider an orbit problem of the automorphism group of F ⋊ t : decide whether an automorphism sends ι(F ) on F and ι(t) in t ′ F .…”

Equation problem over central extensions of hyperbolic groups

“…Rips and Sela [12] reduce the Equation Problem in torsion free hyperbolic groups to the Equation Problem in free groups. Dahmani and Guirardel [3] reduce the problem in hyperbolic groups (possibly with torsion) to solving equations with rational constraints (See Def 2.5) over virtually free groups. Diekert and Muscholl [4] reduce the Equation Problem in right-angled Artin groups to free groups.…”

Decision problems for 3-manifolds and their fundamental groups

“…It generalizes both the word problem and the conjugacy problem, and has been solved for torsion-free hyperbolic groups by Makanin [62] and Rips and Sela [95], for hyperbolic groups with torsion by Dahmani and Guirardel [21] and for fundamental groups of Seifert fibered spaces by Liang [60]. The following question thus arises.…”