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23
Unsolvable problems about small cancellation and word hyperbolic groups
 BULL. LONDON MATH. SOC
, 1994
"... We apply a construction of Rips to show that a number of algorithmic problems concerning certain small cancellation groups and, in particular, word hyperbolic groups, are recursively unsolvable. Given any integer k> 2, there is no algorithm to determine whether or not any small cancellation group ..."
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Cited by 27 (3 self)
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We apply a construction of Rips to show that a number of algorithmic problems concerning certain small cancellation groups and, in particular, word hyperbolic groups, are recursively unsolvable. Given any integer k> 2, there is no algorithm to determine whether or not any small cancellation group can be generated by either two elements or more than k elements. There is a small cancellation group E such that there is no algorithm to determine whether or not any finitely generated subgroup of E is all of E, or is finitely presented, or has a finitely generated second integral homology group.
The isomorphism problem for torsionfree abelian groups is analytic complete
 JOURNAL OF ALGEBRA
, 2008
"... We prove that the isomorphism problem for torsionfree Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete. ..."
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Cited by 13 (6 self)
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We prove that the isomorphism problem for torsionfree Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete.
Conjugacy in normal subgroups of hyperbolic groups
, 2009
"... Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund’s and D. Wise’s version of Rips’ ..."
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Cited by 11 (1 self)
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Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund’s and D. Wise’s version of Rips’s construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non(conjugacy separable) subgroups of finite index.
Rips construction and Kazhdan property
 T), Groups, Geom., & Dynam
, 2006
"... Abstract. We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1→N → G→Q→1, where G is a hyperbolic group and N is a quotient group of H. As an application we construct a representation rigid but not superrigid hyperbolic grou ..."
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Cited by 10 (2 self)
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Abstract. We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1→N → G→Q→1, where G is a hyperbolic group and N is a quotient group of H. As an application we construct a representation rigid but not superrigid hyperbolic group, show that adding relations of the form x n = 1 to a presentation of a hyperbolic group may drastically change the group even in case n>> 1, and prove that some properties (e.g. properties (T) and FA) are not recursively recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier–Wise on outer automorphism groups of Kazhdan groups. 1. Introduction and
Asymptotic invariants, complexity of groups and related problems
 Bulletin of Mathematical Sciences
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Algorithmic Problems in Amalgams of Finite Groups
 Conjugacy and Intersection Properties, arXiv.org: math.GR/0707.0165
"... Abstract. Geometric methods proposed by Stallings [53] for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their ..."
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Cited by 4 (2 self)
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Abstract. Geometric methods proposed by Stallings [53] for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their
ASPHERICAL MANIFOLDS, RELATIVE HYPERBOLICITY, SIMPLICIAL VOLUME, AND ASSEMBLY MAPS
, 2005
"... Abstract. This paper contains examples of closed aspherical manifolds obtained as a byproduct of recent work by the author [Bel] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated nmanifold M n with hyperbolic fundamental group is a ..."
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Abstract. This paper contains examples of closed aspherical manifolds obtained as a byproduct of recent work by the author [Bel] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated nmanifold M n with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n + 1)manifold N n+1 with hyperbolic fundamental group. (II) If B1,... Bm are closed aspherical triangulated nmanifolds, then there is a closed aspherical triangulated manifold N of dimension n + 1 such that N has nonzero simplicial volume, N retracts to each Bk, and π1(N) is hyperbolic relative to π1(Bk)’s. (III) Any finite aspherical simplicial complex is a retract of a closed aspherical triangulated manifold with positive simplicial volume and nonelementary relatively hyperbolic fundamental group. 1.
The triviality problem for profinite completions
, 2013
"... Abstract. We prove that there is no algorithm that can determine whether or not a finitely presented group has a nontrivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, nonpositively curved square complexes. We deduce that many other properties of gr ..."
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Cited by 3 (2 self)
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Abstract. We prove that there is no algorithm that can determine whether or not a finitely presented group has a nontrivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, nonpositively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion. 1.
THE HURWITZ EQUIVALENCE PROBLEM IS UNDECIDABLE
, 2005
"... Abstract. In this paper, we prove that the Hurwitz equivalence problem for 1factorizations in F2⊕F2 is undecidable, and as a consequence, the Hurwitz equivalence problem for ∆ 2factorizations in the braid groups Bn, n ≥ 5 is also undecidable. ..."
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Cited by 1 (1 self)
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Abstract. In this paper, we prove that the Hurwitz equivalence problem for 1factorizations in F2⊕F2 is undecidable, and as a consequence, the Hurwitz equivalence problem for ∆ 2factorizations in the braid groups Bn, n ≥ 5 is also undecidable.
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.