Results 1  10
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35
Connectivity properties of group actions on nonpositively curved spaces II: The geometric invariants
, 1998
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Quasi–actions on trees I, bounded valence
 Annals of Mathematics
"... Given a bounded valence, bushy tree T, we prove that any quasiaction of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T ′. This theorem has many applications: quasiisometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups fo ..."
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Cited by 9 (3 self)
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Given a bounded valence, bushy tree T, we prove that any quasiaction of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T ′. This theorem has many applications: quasiisometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan’s Theorem about uniformly quasiconformal actions on the 2sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasiisometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries. 1
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 8 (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
A Note On Cohomological Vanishing And The Linear Isoperimetric Inequality
, 1996
"... If G is a finitely presented group, then H 2 (#) (G, A) vanishes for all injective Banach spaces iff the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic iff the homological area function satisfi ..."
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Cited by 6 (6 self)
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If G is a finitely presented group, then H 2 (#) (G, A) vanishes for all injective Banach spaces iff the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic iff the homological area function satisfies the linear isoperimetric inequality. A closed 3manifold group G is hyperbolic iff H 2 (#) (G, A) vanishes for all injective Banach spaces A. A vanishing theorem is proved for the fundamental group of a closed Riemannian manifold of negative curvature.
The lower algebraic Ktheory of Fuchsian groups
, 1998
"... Abstract. Let Γ be a cocompact Fuchsian group. We calculate the lower algebraic Ktheory of the integral group ring ZΓ and find an explicit formula for Ki(ZΓ), i ≤ 1, in terms of the lower Kgroups of finite cyclic groups. 1. ..."
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Cited by 5 (1 self)
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Abstract. Let Γ be a cocompact Fuchsian group. We calculate the lower algebraic Ktheory of the integral group ring ZΓ and find an explicit formula for Ki(ZΓ), i ≤ 1, in terms of the lower Kgroups of finite cyclic groups. 1.
Systoles of 2complexes, Reeb graph, and Grushko decomposition
, 2006
"... Let X be a finite 2complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2complexes. Our inequality improves the constan ..."
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Cited by 4 (4 self)
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Let X be a finite 2complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2complexes. Our inequality improves the constant in M. Gromov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model ” for X, reminiscent of the “chopping off long fingers ” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the LusternikSchnirelmann and systolic categories of a 2complex.
Closed essential surfaces in hyperbolizable acylindrical 3manifolds
 Pacific J. Math
, 1998
"... We show that a compact hyperbolizable acylindrical 3manifold with nonempty incompressible boundary, in which every boundary component has genus at least two, necessarily contains a closed immersed essential surface. 1. Introduction. The purpose of this note is to demonstrate the existence of close ..."
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Cited by 3 (2 self)
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We show that a compact hyperbolizable acylindrical 3manifold with nonempty incompressible boundary, in which every boundary component has genus at least two, necessarily contains a closed immersed essential surface. 1. Introduction. The purpose of this note is to demonstrate the existence of closed immersed essential surfaces in a certain class of hyperbolizable 3manifolds. We state here our main result. Theorem 4.2. Let M be a compact hyperbolizable acylindrical 3manifold
A characterisation of virtually free groups
 Arch. Math
"... Abstract. We prove that a finitely generated group G is virtually free if and only if there exists a generating set for G and k> 0 such that all klocally geodesic words with respect to that generating set are geodesic. ..."
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Cited by 2 (1 self)
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Abstract. We prove that a finitely generated group G is virtually free if and only if there exists a generating set for G and k> 0 such that all klocally geodesic words with respect to that generating set are geodesic.
Subgroups of the direct product of two limit groups
, 2005
"... If Γ1 and Γ2 are limit groups and S ⊂ Γ1 × Γ2 is of type FP2 then S has a subgroup of finite index that is a product of at most two limit groups. ..."
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Cited by 2 (2 self)
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If Γ1 and Γ2 are limit groups and S ⊂ Γ1 × Γ2 is of type FP2 then S has a subgroup of finite index that is a product of at most two limit groups.