Research Statement Justin A. James Decision Problems in Group Theory 1 Introduction In 1911, Dehn formulated three fundamental decision problems for groups: the word problem, the conjugacy problem, and the isomorphism problem [Deh]. Dehn proved that there are algorithms to decide each of these problems when G is the fundamental group of a closed 2-manifold. However, these problems are undecidable for finitely presented groups in general. The word problem for a group G given by a finite presentation G= A R asks if there is an algorithm which given any word (a finite length product in the generators and their inverses) determines whether or not the word represents the identity element in the group. Novikov in 1955 [Nov] and Boone in 1959 [Boo] independently proved that there is a finitely presented group with undecidable word problem. Since then, there has been much progress toward the goal of determining which classes of groups have decidable word problem and which have undecidable word problem, but many open questions remain. If we restrict our attention to finitely presented groups that have decidable word problem, there are several generalizations of the word problem one can consider (see [McC] and [LM]). These include: deciding membership in finitely generated subgroups (the generalized word problem), deciding membership in finitely generated submonoids (the submonoid membership problem), determining the order of an element of the group (the order problem), deciding whether or not one element of the group is a non-negative power of another element (the power problem), and deciding membership in rational subsets of the group (the rational membership problem). The rational subsets of a group are subsets built up from singleton sets by recursively applying the operations union, product, and monoid closure. Each of the generalizations of the word problem mentioned above are rational problems because they can be solved by deciding instances of the rational membership problem. These problems are known to be strictly harder than the word problem. That is, an algorithm deciding any one of these problems can also be used to decide the word problem. For each of these decision problems, there is an example of a finitely presented group for which the word problem is decidable and the given problem is undecidable. However, there are examples of specific classes of groups for which these problems can be decided. For instance, it is known that all of these problems are decidable for finitely generated free groups [Ben] and for finitely generated free abelian groups [Gru]. The generalized word problem for finitely generated free abelian groups is of particular interest, because it is known to be an NP-complete problem (it has been reduced to non-negative integer programming, see [Sah]). I have found an alternate algorithm to decide the generalized word problem for finitely generated free abelian groups which only requires basic properties of vectors, and trigonometry. It remains to be seen whether my approach offers a significant improvement in complexity as compared to other existing algorithms.
Justin A. James 2 2 Closure Properties A major theme in group theory is the idea of building new groups from old ones. This can be done using several standard constructions. These include subgroups, quotients, direct products, free products, free products with amalgamation, and HNN extensions. It is natural to ask what effect these constructions have on the decidability of algorithmic problems such as the word problem and its generalizations. A Theorem of Grunschlag [Gru] shows that the decidability of the word problem, the generalized word problem, and the rational membership problem are all preserved for finite index subgroups and finite index extensions of a finitely generated group. K. A. Mihailova has shown that the generalized word problem is not preserved under direct product [Mih1], but that it is preserved under free product [Mih2]. That is, there is an example of two groups with decidable generalized word problem whose direct product has undecidable word problem, while if two finitely generated groups have decidable generalized word problem, then their free product also has decidable generalized word problem. I have been able to extend Mihailova s techniques to prove the following result: Theorem 1: Let G= G 1 G 2 where G 1 and G 2 are groups with generating sets A and B respectively. If G 1 and G 2 have decidable Rational Membership Problem, then membership in finitely generated submonoids of G is decidable. The algorithm used to prove this result is a two step process which starts by constructing a finite collection of finite state automata, and then uses them to perform a Nielsen -like completion procedure (See [LS]) in order to obtain a generating set for which every element of the given submonoid has a well behaved factorization. Finite state automata can be thought of as directed labeled graphs with a distinguished start vertex and a set of distinguished terminal vertices. The language accepted by a finite state automaton is the set of all strings that label a directed path from the start vertex to one of the terminal vertices (See [HU]). If the labels of the edges correspond to a finite set of generators for a group, then the language of an automaton corresponds to a subset of this group - namely the elements of the group that occur as the product of the generators given by a word labeling an accepted path in the automaton. Kleene s Theorem gives a connection between languages accepted by finite state automata labeled by a finite set A and rational subsets of the free monoid A (the set of all finite length strings over the alphabet A). If A is taken to be a finite generating set for a group G, a theorem of Berstel [Ber] shows that the image of a rational subset of a free monoid A is a rational subset of the group G under the natural identification map. Because of this, finite state automata are a computational tool that are often used to decide rational problems in finitely generated groups. I would like to determine whether or not the hypothesis in the previous theorem that the factors have decidable rational membership problem can be weakened. This leads to the following question: Question 1: Let G= G 1 G 2 where G 1 and G 2 are groups with generating sets A and B respectively. If G 1 and G 2 have decidable submonoid membership problem, is membership in finitely generated submonoids of G is decidable? Ben Steinberg has recently shown me an unpublished proof that the rational membership problem is also preserved under free products and free products with amalgamation and HNN extensions with finite base. I plan to investigate extending Steinberg s results to larger classes of amalgams and HNN extensions. 2
Justin A. James 3 3 Word Hyperbolic Groups For any group G with finite generating set A G with A closed under inversion, the Cayley graph of G with respect to A, denotedγ(g, A), is defined as follows: the vertex set of the graph is G, and for each a Aand g G, there is an edge from the vertex g to the vertex ga. Γ(G, A) can be made into a geodesic metric space by taking each edge as isometric to the unit interval, and then using the path metric (the distance between any two points in the graph is the length of a shortest path between them). Cayley graphs allow us to apply topological, combinatorial, and analytical methods to finitely generated groups. In any geodesic metric space X, a geodesic triangle (x, y, z) is calledδ-slim forδ 0if each of its sides is contained in theδ-neighborhood of the union of the other two sides. A geodesic metric space X called δ-hyperbolic if every geodesic triangle is δ-slim. A group G is δ-hyperbolic with respect to a generating set A if its Cayley graph with respect A,Γ(G, A) with path metric is aδ-hyperbolic metric space. G is word hyperbolic if it isδ-hyperbolic for someδ 0and some finite generating set A. Word hyperbolic groups were introduced by Gromov [Gro]. It is known that word hyperbolic groups have decidable word problem [AB], decidable order problem ([Lys] or [Bra]), and decidable power problem [Lys]. However, Rips [Rip] showed that the generalized word problem is undecidable for word hyperbolic groups. Therefore, membership in finitely generated submonoids and rational subsets are also be undecidable. The generalized power problem for a finitely generated groups asks if there is an algorithm that, given two elements of the group via two words u and v, can decide whether or not there are positive integer exponents k and l such that u k and v l represent the same element of the group. This decision problem is strictly harder than the power problem (and hence the word problem as well) and it turns out to be strictly easier than the generalized word problem. By analyzing the geometric properties of geodesic rays and quasigeodesic rays in the Cayley graph of a word hyperbolic group, I have been able to prove the following results: Theorem 2: Let G= A R be a finitely generatedδ-hyperbolic group. Then there is an algorithm deciding the generalized power problem for G. Theorem 3: Let G= A R be a finitely generatedδ-hyperbolic group. Let u, v A. Then there is an algorithm deciding membership in the rational subset{u} {v}. The fact that the generalized word problem is undecidable for word hyperbolic groups in general gives an upper bound on how far we can generalize this result, but I plan to continue to work on determining whether are not there are solutions to the decision problems that lie between the generalized power problem and the generalized word problem. Question 2: Let G= A R be a finitely generatedδ-hyperbolic group. Let u 1, u 2, u 3 A. Is there an algorithm deciding membership in the rational subset{u 1 } {u 2 } {u 3 }? If u 1, u 2,..., u n A, is there an algorithm to decide membership in{u 1 } {u 2 }... {u n }? 3
Justin A. James 4 4 Future Work Beyond the results mentioned above, very little is known about the decidability of generalizations of the word problem for finitely generated groups. There are many interesting classes of groups for which the decidability of rational problems is still open. Two classes of groups for which there are some partial results are Surface groups, and Coxeter groups. The word problem is known to be decidable for both of these classes of groups. Both P. Schupp [Sch] and McCammond and Wise [MW] have applied perimeter reduction techniques to decide the generalized word problem for Coxeter groups when restrictions are put on the both the number of generators of the group, and the exponents of the Coxeter relations. I plan to apply the techniques I have developed to investigate the following questions: Question 3: Is the rational membership problem, the submonoid membership problem, or the generalized word problem decidable for Coxeter groups of extra large type? Question 4: Is the rational membership problem or the submonoid membership problem decidable for surface groups? References [1] [AB] J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short. Notes on word hyperbolic groups (ed. H. Short). In E. Ghys, A. Haefliger, and A. Verjovsky, editors, Group Theory from a Geometrical Viewpoint, pp.3-63. World Scientific, 1991. [2] [Ben] M. Benois. Partie rationelles du groupes libre., C. R. Acad. Sci. Paris, Ser. A, 269:1188-1190, 1969. [3] [Ber] J. Berstel. Transductions and Context-Free Languages, Teubner, Stuttgart, 1979. [4] [Boo] W. W. Boone. Certain simple unsolvable problems in group theory, I, II, III, IV, V VI, Nederl. Akad. Wetensch Proc. Ser A 57, 231-237, 492-497 (1954), 58, 252-256, 571-577 (1955), 60, 22-27, 227-232 (1957). [5] [Bra] N. Brady. Finite Subgroups of Hyperbolic Groups Internat. J. Algebra Comput., 10(2000), no. 4, 399-405. [6] [BrHa] M. R. Bridson and A. Haefliger. Metric Spaces of Non-Positive Curvature, Springer-Verlag, 1999. [7] [Deh] M. Dehn. Über unendliche diskontinuerliche Gruppen, Math. Ann. 69, 116-144, 1911. [8] [Gro] M. Gromov. Hyperbolic Groups, Essays in Group Theory, MSRI series vol. 8, edited by S. M. Gersten, pp. 75-263, Springer Verlag, 1987. [9] [Gru] Z. Grunschlag. Algorithms in Geometric Group Theory, Ph.D. Dissertation, Univ. Calif. Berkeley, 1999. [10] [HU] J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages, and Computation, Adison-Wesley Publishing Co., Reading MA - Menlo Park CA, 1979. 4
Justin A. James 5 [11] [LM] S. Lipschutz and C. F. Miller III. Groups with certain solvable and unsolvable decision problems. Comm. in Pure and Applied Math. 24, 7-15, 1971. [12] [LS] R. C. Lyndon, P. E. Schupp. Combinatorial Group Theory, Springer, Berlin, 1977. [13] [MW] J. P. McCammond and D. T. Wise. Coherence, local quasiconvexity and the perimeter of 2-complexes, Preprint, Texas A&M University, 2001. [14] [McC] J. McCool. The order problem and the power problem for free product sixth-groups. Glascow Math. J. 10, 1-9, 1969. [15] [Mih1] K. A. Mihailova. The occurrence problem for direct products of groups. (English Translation) Math USSR-Sbornik 70, 241-251, 1966. [16] [Mih4] K. A. Mihailova. The occurrence problem for free products of groups. (English Translation) Math USSR-Sbornik 75, 181-190, 1968. [17] [Nov] P. S. Novikov. On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44, 1-143, 1955. [18] [Rip] E. Rips. Subgroups of small cancellation groups. Bull. London Math. Soc. 14, 45-47, 1982. [19] [Sah] S. Sahni, Computationally Related Problems. SIAM J. Comput, Vol. 3, No. 4, pp. 262-279, 1974. [20] [Sch] P. E. Schupp. Coxeter groups, 2-completion, perimeter reduction and subgroup separability. Geom. Dedicata 96, 179-198, 2003. 5