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54
Hyperbolic Groups With Low Dimensional Boundary
 Ann. Sci. École Norm. Sup
, 2000
"... If a torsionfree hyperbolic group G has 1dimensional boundary @1G, then @1G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3dimensional hyperbolic Poincar'e duality group. We ..."
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Cited by 31 (10 self)
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If a torsionfree hyperbolic group G has 1dimensional boundary @1G, then @1G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3dimensional hyperbolic Poincar'e duality group. We also construct a "topologically rigid" hyperbolic group G: any homeomorphism of @1G is induced by an element of G.
Subgroups of word hyperbolic groups in dimension 2
 Jour. London Math. Soc
, 1996
"... If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbol ..."
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Cited by 22 (10 self)
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If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are denned for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to 'relative hyperbolicity'. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range. 1.
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 ca ..."
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Cited by 15 (2 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.
Coherence, local quasiconvexity and the perimeter of 2complexes
, 2002
"... A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2complexes which is introduced here. ..."
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Cited by 15 (4 self)
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A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that onerelator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3manifold groups. The main application is to establish the coherence
Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 13 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
A Cohomological Characterization Of Hyperbolic Groups
, 1996
"... A finitely presented group G is word hyperbolic i# H 2 (#) (G, ## ) = 0. ..."
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Cited by 13 (4 self)
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A finitely presented group G is word hyperbolic i# H 2 (#) (G, ## ) = 0.
Kazhdan groups with infinite outer automorphism group
 Trans. Amer. Math. Soc., ArXiv math.GR/0409203
"... Abstract. For each countable group Q we produce a short exact sequence 1 → N → G → Q → 1 where G has a graphical 1 6 presentation and N is f.g. and satisfies property T. As a consequence we produce a group N with property T such that Out(N) is infinite. Using the tools developed we are also able to ..."
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Cited by 11 (2 self)
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Abstract. For each countable group Q we produce a short exact sequence 1 → N → G → Q → 1 where G has a graphical 1 6 presentation and N is f.g. and satisfies property T. As a consequence we produce a group N with property T such that Out(N) is infinite. Using the tools developed we are also able to produce examples of nonHopfian and noncoHopfian groups with property T. One of our main tools is the use of random groups to achieve certain properties. 1.
Finitely presented subgroups of automatic groups and their isoperimetric functions
 J. London Math. Soc
, 1997
"... Abstract. We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgr ..."
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Cited by 10 (0 self)
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Abstract. We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality. 1. Introduction. Despite
and R.Weidmann, Freely indecomposable groups acting on hyperbolic spaces
 Internat. J. Algebra Comput
"... Abstract. We obtain a number of finiteness results for groups acting on Gromovhyperbolic spaces. In particular we show that a torsionfree locally quasiconvex hyperbolic group has only finitely many conjugacy classes of ngenerated oneended subgroups. 1. ..."
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Cited by 10 (8 self)
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Abstract. We obtain a number of finiteness results for groups acting on Gromovhyperbolic spaces. In particular we show that a torsionfree locally quasiconvex hyperbolic group has only finitely many conjugacy classes of ngenerated oneended subgroups. 1.
van Kampen’s embedding obstructions for discrete groups
 Invent. Math
, 2002
"... We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1 ..."
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Cited by 9 (2 self)
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We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1