DIOPHANTINE GEOMETRY OVER GROUPS:

A LIST OF RESEARCH PROBLEMS

The research problems below are all related to the work on Diophantine geometry over groups. We intend to update the list periodically. Comments, additional problems, and (particularly) solutions, are welcome.

Part I: Limit groups (ω-residually free groups)

(1) Let L be a limit group. Conjecture:

(i) L is hyperbolic relative to its maximal non-cyclic abelian subgroups. (ii) L is CAT(0). Furthermore, L is CAT(0) with isolated flats (as defined

by C. Hruska and D. Wise).

(2) Conjecture: a f.g. group is a limit group if and only if it acts freely on an R^{n} tree for some positive integer n.

(3) Give sufficient or necessary conditions for a cyclically pinched group to be a limit group. Note that a double of a free group along a primitive element is a limit group, but there are cyclically pinched groups that are not limit groups, e.g., F ∗_{<w_{1}=w_{2}>} F where the element w_{1} is a commutator in the first free group and w_{2} is a product of two ”high” powers in the other free group.

(4) Classify limit groups of small rank.

(5) Conjecture: the conjugacy and isomorphism problems for limit groups are decidable.

(6) Is the generalized word problem decidable for a limit group?

(7) Can one decide if a given f.g. subgroup of a limit group is malnormal? Note that the answer is positive for free groups [BMR], but negative for general hyperbolic groups [Br-Wi].

(8) Let G be a CAT(0) group with isolated flats (as defined by C. Hruska and

D. Wise).

(i) Is G Hopf? (ii) Is it true that every system of equations over G is equivalent to a finite

system? (iii) Is it possible to associate a Makanin-Razborov diagram that encodes

the entire set of solutions to a system of equations defined over G, i.e., to generalize the Makanin-Razborov diagram associated with a system of equations defined over a hyperbolic group?

(iv) Is it possible to decide if a system of equations over G has a solution?

(9) For each positive integer n, let:

x^{ϵ_{1}}p_{1} … x^{ϵ_{n}}p_{n} = 1

be a parametric equation, where ϵ_{i} = ±1, for 1 ≤ i ≤ n. For each n and each value of the parameters p_{1}, …, p_{n}, the system has at most s(n) ”sporadic” solutions, and at most c(n) sets of solutions of the form x = b_{1}w^{λ}b_{2}, where λ is an integer. Can one give explicit bounds on the functions s(n) and c(n)? Can one give an ”asymptotic description” of the graded Makanin-Razborov diagram?

(10) For each positive integer g, let:

p = [x_{1}, x_{2}] … [x_{2g−1}, x_{2g}]

be a parametric equation over some free group F_{k}. For each element p_{0} ∈ F_{k}, which is an element of genus g, the equation obtained from the parametric equation by setting p = p_{0} consists of f(p_{0}) families of solutions, where each family is an orbit of the action of the modular group of a once-punctured surface of genus g on the set of solutions. Furthermore, by [Se3], there exists a global bound c(g) that bounds the number of families f(p_{0}) for all elements p_{0} ∈ F_{k} which are of genus g. Can one estimate the function c(g)?

(11) More generally, Can one estimate the ”size” (or ”complexity”) of the graded Makanin-Razborov diagram of a parametric system of equations in terms of the (combinatorial) complexity of the parametric system?

(12) Can one generalize the result of [BHMS] to arbitrary limit groups? i.e., let L_{1}, …, L_{n} be limit groups. Is it true that every F P_{n} subgroup of the direct product L_{1} + … + L_{n} is virtually a direct product of at most n limit groups which are subgroups of L_{1}, …, L_{n}?

(13) (Delzant) Is the set of limit groups recursively enumerable? Is it recursive? What about the set of residually-free groups?

(14) Are ω-residually free towers ([Se1],6) LERF? Are all limit groups LERF?

Part II: Residually free (hyperbolic) groups

Let H be a residually-free group. H is a semi-direct product of ω-residually free groups, an ω-residually free group is f.p., hence, a residually-free group is recursively presented. Let G be a f.g. group. Let K be the intersection of the kernels of all possible homomorphisms h : G → F, where F is a free group. Let RF (G), the residual free quotient of G, be the quotient RF (G) = G/K. RF (G) is naturally residuallyfree.

(1) Let H be a residually-free group. By Makanin, the word problem in H is decidable. Is the conjugacy problem in H decidable? Is the isomorphism problem for residually-free groups decidable? Are these (and the word) problems decidable for a residually (torsion-free) hyperbolic group?

(2) (Rips) Does the Kervaire conjecture hold in the category of residually-free groups? i.e., let H be a residually free group and let:

G =< H, t_{1}, …, t_{m} | r_{1}, …, r_{m} >

where the the determinant of the matrix corresponding to the occurrences of the generators t_{i} in the relations r_{j} is ±1. Is the natural map ν : H → RF (G) an embedding?

(3) Let G =< t_{1}, …, t_{m} | r_{1}, …, r_{m} > be a f.p. group with a balanced presentation, and suppose that RF (G) is trivial. Is there any analogue of the Andrews-Curtis conjecture for this case? i.e., is there a natural way to transform the relations into the the obvious ones, keeping the triviality of the residually-free quotient? Note that one can always apply Makanin’s machine, but this uses fractions of the variables. We look for transformations of the relations in the free group generated by < t_{1}, …, t_{m} > in the special case in which RF (G) is trivial and the presentation is balanced.

(4) (Potyagailo) What 3-manifold groups are residually-free? what cyclically pinched ones?

Part III: The elementary theory of (free) groups

(1) Conjecture: the elementary theory of a (torsion-free) hyperbolic group is stable.

(2) Conjecture: there exists an elementary set defined over a free group F that is not an AE set.

(3) (Rips) Let H be an elementary subgroup of a free group F. Conjecture: H is either cyclic or the entire free group F.

(4) Is it true that the only definable subgroups of a hyperbolic group are either virtually cyclic or the entire group?

(5) (Rips) Let F_{n} be a free group of rank n ≥ 3. Let F B_{n} ⊂ F_{n}^{n} be the set of

free bases of F_{n}. Conjecture: F B_{n} is not an elementary set.

Note that F B_{2} is elementary by a theorem of Malcev:

F B_{2}(x, y) = ∃w w[x, y]w^{−1} = [a_{1}, a_{2}]

where < a_{1}, a_{2} >= F_{2}. Also, note that it is enough to show that F B_{n} is not in the Boolean algebra of AE sets.

(6) Can one classify those (torsion-free) hyperbolic groups, for which the set of their generating systems of fixed cardinality is definable?

(7) Is the universal theory of a (torsion-free) hyperbolic group decidable? Is the elementary theory of a (torsion-free) hyperbolic group decidable? Note that the Diophantine theory of torsion-free hyperbolic groups is known to be decidable.

(8) Let G_{1}, G_{2} be (torsion-free) CAT(0) groups with isolated flats. Is it true that G_{1} is elementary equivalent to G_{2} if and only if G_{1} is AE equivalent to G_{2}? Is it true that every elementary set defined over such a group is in the Boolean algebra of AE sets?

(9) Can one classify those f.g. groups that are elementary equivalent to a lattice in a semi-simple Lie group? Note that the answer is positive for uniform lattices in rank 1 Lie groups.

(10) (Sabbagh) Let G_{1}, G_{2}, H_{1}, H_{2} be f.p. groups, and suppose that G_{1} is elementary equivalent to H_{1}, and G_{2} is elementary equivalent to H_{2}. Is G_{1}∗G_{2} elementary equivalent to H_{1} ∗ H_{2}? Note that the answer is positive if both H_{1} and H_{2} are torsion-free hyperbolic.

(11) (Rips) In [Se7] we associate an elementary core with any given torsion-free hyperbolic group G, EC(G), and prove that two torsion-free hyperbolic groups are elementary equivalent if and only if their elementary cores are isomorphic. The elementary core of a (torsion-free) hyperbolic group G is a retract of the ambient hyperbolic group, and it is defined iteratively. At each step of the construction we factor the obtained group into a free product, associate with each factor its cyclic JSJ decomposition and then look for a possible retract of a particular type that is associated with the JSJ, with which we continue to the next step of the construction (see section 3 of [Se7]). The construction is shown to terminate after finitely many steps, and the isomorphism type of the obtained core does not depend on the particular procedure that was used for its construction.

(i) Clearly, the construction of the elementary core can be generalized to an arbitrary f.p. group. Does it always terminate after finitely many steps?

(ii) If it does terminate, is the isomorphism type of the obtained core an invariant of the ambient group? (i.e., is it true that the isomorphism type of the core does not depend on the particular procedure that was used for its construction)

References

[BMR] G. Baumslag, A. Myasnikov and V. Remeslennikov, Malnormality is decidable in free groups, Int. jour. of algebra and comp., to appear.

[BHMS] M. Bridson, J. Howie, C. Miller and H. Short, The subgroups of direct products of surface groups, preprint.

[Br-Wi] M. Bridson and D. Wise, Malnormality is undecidable in hyperbolic groups, Israel journal of Math, to appear.

[Se1-Se7] Z. Sela, Diophantine geometry over groups I-VII, preprints.