Results 1  10
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48
Relatively hyperbolic groups
 Michigan Math. J
, 1998
"... Abstract. We generalize some results of Paulin and RipsSela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not coHopfian or Out(G) ..."
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Cited by 32 (2 self)
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Abstract. We generalize some results of Paulin and RipsSela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not coHopfian or Out(G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric Haction on an Rtree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of OllivierWise). 1.
Lpresentations and branch groups
, 2002
"... Abstract. We introduce Lpresentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations. 1. ..."
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Cited by 28 (9 self)
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Abstract. We introduce Lpresentations: group presentations endowed with a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations. 1.
Ideal bicombings for hyperbolic groups and applications
 Topology
"... Abstract. For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orb ..."
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Cited by 18 (4 self)
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Abstract. For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [MSb] hold for all nonelementary hyperbolic groups and their nonelementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces. 1.
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
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Cited by 17 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
Regular neighbourhoods and canonical decompositions for groups
, 2008
"... We find canonical decompositions for finitely presented groups which specialize to the classical JSJdecomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept o ..."
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Cited by 15 (2 self)
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We find canonical decompositions for finitely presented groups which specialize to the classical JSJdecomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood for a finite family of almost invariant subsets of a group. An almost invariant set is an analogue of an immersion.
Quasiactions on trees II: Finite depth BassSerre trees
, 2004
"... This paper addresses questions of quasiisometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the BassSerre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge grou ..."
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Cited by 14 (2 self)
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This paper addresses questions of quasiisometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the BassSerre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincare duality groups. The main theorem says that, under certain hypotheses, if G is a finite graph of coarse Poincare duality groups then any finitely generated group quasiisometric to the fundamental group of G is also the fundamental group of a finite graph of coarse Poincare duality groups, and any quasiisometry between two such groups must coarsely preserves the vertex and edge spaces of their BassSerre trees of spaces. Besides some simple normalization hypotheses, the main hypothesis is the “crossing graph condition”, which is imposed on each vertex group Gv which is an ndimensional coarse Poincare duality group for which every incident edge group has positive codimension: the crossing graph of Gv is a graph ǫv that describes the pattern in which the codimension 1 edge groups incident to Gv are crossed by other edge groups incident to Gv, and the crossing graph condition requires that ǫv be connected or empty. 1
Endomorphic presentations of branch groups
 J.Algebra
"... Abstract. We introduce “endomorphic presentations”, or Lpresentations: group presentations whose relations are iterated under a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations, generalisin ..."
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Cited by 13 (3 self)
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Abstract. We introduce “endomorphic presentations”, or Lpresentations: group presentations whose relations are iterated under a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite Lpresentations, generalising results by Igor Lysionok and Said Sidki. 1.
Decision problems in group theory
 Proc. London Math. Soc
, 1982
"... At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an ex ..."
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Cited by 10 (0 self)
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At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an exposition of modular