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30
The Isomorphism Problem for Toral Relatively Hyperbolic Groups
"... We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."
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Cited by 40 (8 self)
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We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic nmanifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsionfree relatively hyperbolic group with abelian parabolics is
Recognizing string graphs in NP
 J. of Computer and System Sciences
"... A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable u ..."
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Cited by 31 (5 self)
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A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph (Pach and Tóth, 2001; Schaefer and ˇ Stefankovič, 2001). These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NPcomplete, since Kratochvíl showed that the recognition problem is NPhard (Kratochvíl, 1991b). The result has consequences for the computational complexity of problems in graph drawing, and topological inference. We also show that the string graph problem is decidable for surfaces of arbitrary genus. Key words: String graphs, NPcompleteness, graph drawing, topological inference, Euler diagrams
Solvability of Equations in Free Partially Commutative Groups Is Decidable
 INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
, 2001
"... Trace monoids are wellstudied objects in computer science where they serve as a basic algebraic tool for analyzing concurrent systems. ..."
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Cited by 20 (6 self)
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Trace monoids are wellstudied objects in computer science where they serve as a basic algebraic tool for analyzing concurrent systems.
Word equations over graph products
 In Proceedings of the 23rd Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2003), Mumbai (India), number 2914 in Lecture Notes in Computer Science
, 2003
"... For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those fac ..."
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Cited by 13 (8 self)
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For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.
CONJUGACY CLASSES OF SOLUTIONS TO EQUATIONS AND INEQUATIONS OVER HYPERBOLIC GROUPS
, 2009
"... We study conjugacy classes of solutions to systems of equations and inequations over torsionfree hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups i ..."
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Cited by 9 (1 self)
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We study conjugacy classes of solutions to systems of equations and inequations over torsionfree hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsionfree hyperbolic group.
Existential and Positive Theories of Equations in Graph Products
 Theory of Computing Systems
, 2003
"... We prove that the existential theory of equations with normalized rational constraints in a fixed graph product of finite monoids, free monoids, and free groups is PSPACEcomplete. Under certain restrictions this result also holds if the graph product is part of the input. As the second main resu ..."
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Cited by 7 (6 self)
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We prove that the existential theory of equations with normalized rational constraints in a fixed graph product of finite monoids, free monoids, and free groups is PSPACEcomplete. Under certain restrictions this result also holds if the graph product is part of the input. As the second main result we prove that the positive theory of equations with recognizable constraints in graph products of finite and free groups is decidable.
Asymptotic invariants, complexity of groups and related problems
 Bulletin of Mathematical Sciences
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Theories of HNNextensions and amalgamated products
 Proceedings of the 33st International Colloquium on Automata, Languages and Programming (ICALP 2006), Venice (Italy), number 4052 in Lecture Notes in Computer Science
, 2006
"... Abstract. It is shown that the existential theory of G with rational constraints, over an HNNextension G = 〈H, t; t −1 at = ϕ(a)(a ∈ A) 〉 is decidable, provided that the same problem is decidable in the base group H and that A is a finite group. The positive theory of G is decidable, provided that ..."
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Cited by 7 (4 self)
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Abstract. It is shown that the existential theory of G with rational constraints, over an HNNextension G = 〈H, t; t −1 at = ϕ(a)(a ∈ A) 〉 is decidable, provided that the same problem is decidable in the base group H and that A is a finite group. The positive theory of G is decidable, provided that the existential positive theory of G is decidable and that A and ϕ(A) are proper subgroups of the base group H with A ∩ ϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtuallyfree group is decidable. 1
ON SYSTEMS OF EQUATIONS OVER FREE PARTIALLY COMMUTATIVE GROUPS
, 2009
"... Using an analogue of MakaninRazborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (rightangled Artin group) G. Equivalently, we give a parametrisation of Hom(G, G), where G is a finitely generated group. ..."
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Cited by 5 (0 self)
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Using an analogue of MakaninRazborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (rightangled Artin group) G. Equivalently, we give a parametrisation of Hom(G, G), where G is a finitely generated group.
Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups
, 2007
"... The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theorie ..."
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Cited by 5 (2 self)
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The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifierfree formulas over a nonabelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary firstorder formulas lift from H to H ∗ F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.