Results 1  10
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18
Asymptotic dimension of relatively hyperbolic groups
, 2004
"... Abstract. Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups {H1,..., Hm}. We prove that if each of the subgroups H1,..., Hm has finite asymptotic dimension, then asymptotic dimension of G is also finite. 1. ..."
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Cited by 29 (2 self)
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Abstract. Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups {H1,..., Hm}. We prove that if each of the subgroups H1,..., Hm has finite asymptotic dimension, then asymptotic dimension of G is also finite. 1.
On homological coherence of discrete groups
 J. Algebra
"... Abstract. We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The new notion is defined in terms of the coarse geometry of groups and should be as useful for computing their Ktheory. We prove that a group Γ of finite asymptotic dimension is weakly coherent. ..."
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Cited by 16 (3 self)
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Abstract. We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The new notion is defined in terms of the coarse geometry of groups and should be as useful for computing their Ktheory. We prove that a group Γ of finite asymptotic dimension is weakly coherent. In particular, there is a large collection of R[Γ]modules of finite homological dimension when R is a finitedimensional regular ring. This class contains wordhyperbolic groups, Coxeter groups and, as we show, the cocompact discrete subgroups of connected Lie groups. 1.
ON THE KTHEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION
"... Abstract. It is proved that the assembly maps in algebraic K and Ltheory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model ..."
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Abstract. It is proved that the assembly maps in algebraic K and Ltheory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model for EΓ. In particular, it applies to discrete subgroups of virtually connected Lie groups.
Cohomological approach to asymptotic dimension
, 2006
"... Abstract. We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdimZ X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of ..."
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Cited by 12 (1 self)
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Abstract. We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdimZ X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension. Gromov proposed to study discrete groups as large scale geometric objects. He introduced several asymptotic invariants of finitely generated groups [Gr1]. Among them there is the notion of asymptotic dimension which proved to be useful for the Novikovtype conjectures [Yu],[Ba],[CG],[Dr2],[DFW]. The asymptotic dimension differs from any
A notion of geometric complexity and its applications to topological rigidity, Invent math DOI
, 2010
"... Abstract. We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×Rn is homeomorphic to ..."
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Cited by 11 (1 self)
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Abstract. We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M×Rn is homeomorphic to N×Rn, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K, all elementary amenable groups, and is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions. 1.
An étale approach to the Novikov Conjecture
 Comm. Pure Appl. Math
"... Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the ..."
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Cited by 11 (7 self)
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Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups. Ten years ago, the most popular approach to the Novikov conjecture went via compactifications. If a compact aspherical manifold, say, has a universal cover which suitably equivariantly compactifies, already Farrell and Hsiang [FH] proved that the Novikov conjecture follows. Subsequent work by many authors weakened
On macroscopic dimension of rationally essential manifolds
 Geom. Topol
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Finite decomposition complexity and the integral Novikov conjecture for higher algebraic Ktheory
, 2013
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DISCRETE GROUPS WITH FINITE DECOMPOSITION COMPLEXITY
"... Abstract. In [GTY] we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a closed aspherical manifold whose universal cover, or equivalently whose fundamental group, h ..."
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Cited by 2 (1 self)
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Abstract. In [GTY] we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a closed aspherical manifold whose universal cover, or equivalently whose fundamental group, has FDC. In this note we continue our study of FDC, focusing on permanence and the relation to other coarse geometric properties. In particular, we prove that the class of FDC groups is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions. As consequences we obtain further examples of FDC groups – all elementary amenable groups and all countable subgroups of almost connected Lie groups have FDC. 1.
Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups
, 2006
"... The original Novikov conjecture concerns the (oriented) homotopy invariance of higher signatures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral ..."
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Cited by 2 (0 self)
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The original Novikov conjecture concerns the (oriented) homotopy invariance of higher signatures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence related Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all Sarithmetic subgroups of reductive linear algebraic groups over number fields. For reductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsionfree Sarithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and Sarithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1