Results 1  10
of
15
A cohomological description of property (T) for quantum groups
 J. Funct. Anal
"... ar ..."
(Show Context)
Homological Invariants and QuasiIsometry
"... ABSTRACT. Building upon work of Y. Shalom we give a homologicalalgebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cdR over a commutative ring R satisfies the inequality cdR(Λ) ≤ cdR(Γ) ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. Building upon work of Y. Shalom we give a homologicalalgebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cdR over a commutative ring R satisfies the inequality cdR(Λ) ≤ cdR(Γ) if Λ embeds uniformly into Γ and cdR(Λ) < ∞ holds. Another consequence of our results is that the Hirsch ranks of quasiisometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasiisometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to NovikovShubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasiisometry. 1.
HIGHERORDER SIGNATURE COCYCLES FOR SUBGROUPS OF MAPPING CLASS GROUPS AND HOMOLOGY CYLINDERS
"... Abstract. We define families of invariants for elements of the mapping class group of Σ, a compact orientable surface. Fix any characteristic subgroup H ⊳ π1(Σ) and restrict to J(H), any subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H is as ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We define families of invariants for elements of the mapping class group of Σ, a compact orientable surface. Fix any characteristic subgroup H ⊳ π1(Σ) and restrict to J(H), any subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H is associated a higherorder ρψinvariant and a signature 2cocycle σψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each ρψ is a quasimorphism and each σψ is a bounded 2cocycle. We make calculations in the simplest nontrivial case and show that, by varying ψ, we get infinite families of linearly independent quasimorphisms and signature cocycles. We introduce new subgroups of the Torelli group and show that the ρψ restrict to homomorphisms on these subgroups. Many of these invariants extend to the full mapping class group and some extend to the monoid of homology cylinders based on Σ. 1.
A HochschildSerre spectral sequence for extensions of discrete measured groupoids, arXiv math.DS (2007Jul), available at arXiv:0707.0906
"... Abstract. We construct a HochschildSerre spectral sequence for L 2type cohomology groups of discrete measured groupoids using a blend of tools from homological algebra and ergodic theory. Applications include a theorem about existence of normal amenable subrelations in measured equivalence relatio ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. We construct a HochschildSerre spectral sequence for L 2type cohomology groups of discrete measured groupoids using a blend of tools from homological algebra and ergodic theory. Applications include a theorem about existence of normal amenable subrelations in measured equivalence relations, results about possible stabilizers of measure preserving group actions, and vanishing results or explicit computations of L 2Betti numbers of groups and manifolds. Contents 1. Introduction and Statement of Results 2 1.1. Applications to ergodic theory 2 1.2. Applications to L2Betti numbers of groups 4 1.3. An application to L2Betti numbers of manifolds 5
A SPECTRAL SEQUENCE TO COMPUTE L 2BETTI NUMBERS OF GROUPS AND GROUPOIDS
, 707
"... Abstract. We construct a spectral sequence for L 2type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the HopfSinger conjecture for aspherical manifolds with polysurface fundamental groups. More generally, we obtain a permanence result for the HopfSing ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We construct a spectral sequence for L 2type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the HopfSinger conjecture for aspherical manifolds with polysurface fundamental groups. More generally, we obtain a permanence result for the HopfSinger conjecture under taking fiber bundles whose base space is an aspherical manifold with polysurface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L 2Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations. 1.
L 2COHOMOLOGY FOR VON NEUMANN ALGEBRAS
, 2006
"... Abstract. We study L 2Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [3]. We give a definition of L 2cohomology and show how the study of the first L 2Betti number can be related with the study of derivations with values in a bimodule of affiliated operato ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We study L 2Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [3]. We give a definition of L 2cohomology and show how the study of the first L 2Betti number can be related with the study of derivations with values in a bimodule of affiliated operators. We show several results about the possibility of extending derivations from subalgebras and about uniqueness of such extensions. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest. 1.
QUASIISOMETRY INVARIANCE OF NOVIKOVSHUBIN INVARIANTS FOR AMENABLE GROUPS
, 2003
"... Abstract. We use the notion of uniform measure equivalence to prove that the NovikovShubin invariants resp. the capacities of amenable groups are invariant under quasiisometry. Further, we comment on the connection to Gaboriau’s theorem on the invariance of L 2Betti numbers under orbit equivalenc ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We use the notion of uniform measure equivalence to prove that the NovikovShubin invariants resp. the capacities of amenable groups are invariant under quasiisometry. Further, we comment on the connection to Gaboriau’s theorem on the invariance of L 2Betti numbers under orbit equivalence. 1.