On isomorphisms to a free group and beyond

This paper is devoted to the first-order theories of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if H H is elementarily embedded in a torsion-free hyperbolic group G G , then G G is a tower over H H relative to H H (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin’s results just mentioned is true. This follows from the solution to Tarski’s problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé’s method, we associate to every torsion-free hyperbolic group H H a unique homogeneous countable group M {\mathcal {M}} in which any hyperbolic group H ′ H’ elementarily equivalent to H H has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group H H are H H -limit groups.

The paper is a part of an ongoing program which aims to show that the problem of satisfiability of a system of equations in a free group (hyperbolic or even toral relatively hyperbolic group) is NP-complete. For that, we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi-linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free δ-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in δ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic groups using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length N there is an SLP for another solution such that the size of the SLP is bounded by some polynomial p(s + log N) where s is the size of the system.

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of denable sets over free and hyperbolic groups. In this eighth paper we use a modication of the sieve procedure, which was used in proving quantier elimination in the theory of a free group, to prove that free and torsion-free (Gromov) hyperbolic groups are stable. In the

We use canonical representatives in hyperbolic groups to reduce the theory of equations in (torsion-free) hyperbolic groups to the theory in free groups. As a result we get an effective procedure to decide if a system of equations in such groups has a solution. For free groups, this question was solved by Makanin [Ma]|and Razborov [Ra]. The case of quadratic equations in hyperbolic groups has already been solved by Lysenok [Ly]. Our whole construction plays an essential role in the solution of the isomorphism problem for (torsion-free) hyperbolic groups ([Se1],[Se2]).

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first-order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent.

Part I. The Type Problem: 1. Basic facts 2. Recurrence and transience of infinite networks 3. Applications to random walks 4. Isoperimetric inequalities 5. Transient subtrees, and the classification of the recurrent quasi transitive graphs 6. More on recurrence Part II. The Spectral Radius: 7. Superharmonic functions and r-recurrence 8. The spectral radius 9. Computing the Green function 10. Spectral radius and strong isoperimetric inequality 11. A lower bound for simple random walk 12. Spectral radius and amenability Part III. The Asymptotic Behaviour of Transition Probabilities: 13. The local central limit theorem on the grid 14. Growth, isoperimetric inequalities, and the asymptotic type of random walk 15. The asymptotic type of random walk on amenable groups 16. Simple random walk on the Sierpinski graphs 17. Local limit theorems on free products 18. Intermezzo 19. Free groups and homogenous trees Part IV. An Introduction to Topological Boundary Theory: 20. Probabilistic approach to the Dirichlet problem, and a class of compactifications 21. Ends of graphs and the Dirichlet problem 22. Hyperbolic groups and graphs 23. The Dirichlet problem for circle packing graphs 24. The construction of the Martin boundary 25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth 27. The Martin boundary of hyperbolic graphs 28. Cartesian products.

Links, pictures and the homology of nilpotent groups

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE(n2 log n) for the torsion-free case and NSPACE(n4 log n) in the torsion case. Furthermore, in the presence of effective quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Jeż, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in $\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and $\mathsf{NSPACE}(n^4\log n)$ in the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Jez, Diekert and others on $\mathsf{PSPACE}$ algorithms to solve equations in free monoids and groups using compression, and an intricate language-theoretic analysis. The present work gives an essentially optimal formal language description for all solutions in all hyperbolic groups, and an explicit and surprising low space complexity to compute them.

It is well known that in the theory of abstract groups an important role is played by the free groups of which the quotient groups make up an inexhaustible resource for the abstract groups. There exist, however, some more vast classes of abstract groups which contain the free groups as very particular case; they are the classes of P-groups. Any abstract multiplicative group can, as we know, be given by a set A = {a i }, i ∈ I, of generators and an exhaustive set F of defining relations which connect these generators. Any relation between the elements of A results from the relations F and the trivial relations between the elements of A. There exist some properties named P-properties which can be common to all relations, including the trivial relations connecting the elements of certain sets of generators of multiplicative groups. At present, we know about thirty such properties, to each of them corresponds a vast class of P-groups which yields to an elegant general theory. Here, we present some important classes of P-groups, especially the quasi free groups, the quasi free groups modulo n, the quasi free groups moduli N, the free groups modulo n, the free groups moduli N and the P-symmetric groups. Next, we define the P-products of groups, products presenting some analogies with the free product, and we indicate some properties of these products. Then, we introduce the fundamental and the quasi fundamental groups as well as their bases and we define the rank, the essential invariant, of a fundamental group. To conclude, we review some curious properties of two fundamental groups of rank 2, the first is given by a couple of generators bound by an exhaustive set of two defining relations and the second is given by a couple of generators bound by a single defining relation.

Free Group and Galois Theory are both new contents of modern mathematics with being widely studied, while there are few studies about the connections between them. We can find that the automorphisms in the definition of Galois Group is like the symbols in free group. The structure of subgroups in free groups is a significant subject which is originated from the Group Theory. Thus, this paper will introduce some basic theorems of Galois Group, free group and give a proof of the set of every subfield of a field is isomorphic to a free Galois Group.

Irreducible Affine Varieties over a Free Group: I. Irreducibility of Quadratic Equations and Nullstellensatz

3-34 (1959). 7. G. E. Wall, "On Hughes' Hp-problem," in: Proc. Intern. Conf. Theory of Groups, Canberra, 1965, Gordon and Breach, New York (1967), pp. 357-362. 8. E. I. Khukhro, "On connection between Hughes' conjecture and relations in free groups of prime period," Mat. Sb., 116, No. 2, 253-264 (1981). 9. E. I. Khukhro, "On the Lie ring of a free 2-generated group of prime period and on Hughes' conjecture for 2-generated p-groups," Mat. Sb., 118, No. 4, 567-575 (1982). I0. E. I. Khukhro, "A new identity in the Lie ring of a free group of prime period and groups without Hughes' property," Izv. Akad. Nauk SSSR, Set. Mat., 50, No. 6, 1308-1325 (1986). ii. G. E. Wall, "On the multilinear identities which hold in the Lie ring of a group of prime-power exponent," J. Algebra, 104, 1-22 (1986). 12. J. R. McMullen, "Vortical pro-p-groups and the Hughes' conjecture," Preprint, Univ. of Sydney (1985). 13. R. Warfield, Nilpotent Groups, Lect. Notes in Math., Vol. 517, Springer-Verlag, Berlin (1976). 14. O. H. Kegel, "Die Nilpotenz der Hp-Gruppen," Math. Z., 75, 373-376 (1961).

In this chapter we consider homotopy theories of simplicial objects which resemble the homotopy theory of simplicial groups. It is well known that in the Quillen model category of simplicial groups ΔGr all objects are fibrant; i.e. all simplicial groups satisfy the Kan extension condition. Moreover the free simplicial groups form a sufficiently large class of cofibrant objects in the sense that the homotopy category of free simplicial groups is equivalent to the homotopy category Ho(ΔGr) defined by localization with respect to weak equivalences. Since free groups are cogroups we see that free simplicial groups are simplicial objects in a special theory T of cogroups. In this chapter we study the homotopy theory of “free” simplicial objects in any theory of cogroups, or more generally in any theory of coactions. Such homotopy theories are canonical generalizations of the homotopy theory of simplicial groups.