Extending Properties to Relatively Hyperbolic Groups

Consider a finitely generated group G that is relatively hyperbolic with respect to a family of subgroups H1,…,Hn. We present an axiomatic approach to the problem of extending metric properties from the subgroups Hi to the full group G. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity

On the Hochschild and cyclic (co)homology of rapid decay group algebras

We show that the technical condition of solvable conjugacy bound, introduced in \cite{JOR1}, can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands HH_*^t(\BG)_{} and HC_*^t(\BG)_{} for any bounding class \B, discrete group with word-length $(G,L)$ and conjugacy class $\in$. We use this description to prove the conjecture \B-SrBC of \cite{JOR1} for a class of groups that goes well beyond the cases considered in that paper. In particular, we show that the conjecture $\ell^1$-SrBC (the Strong Bass Conjecture for the topological $K$-theory of $\ell^1(G)$) is true for all semihyperbolic groups which satisfy SrBC, a statement consistent with the rationalized Bost conjecture for such groups

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type $HF^\infty$, i.e. that has a classifying space with the homotopy type of a cellular complex with finitely many cells in each dimension, we show that the isocohomological property is equivalent to the universal cover of the classifying space satisfying polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable (respectively $HF^\infty$ and isocohomological), then we show that the group itself has these respective properties too. Combining with the results of Connes-Moscovici and Dru{\c{t}}u-Sapir we conclude that a group satisfies the Novikov conjecture if it is relatively hyperbolic to subgroups that are of property RD, of type $HF^\infty$ and isocohomological.Comment: 35 pages, no figure

Relative property A and relative amenability for countable groups

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a group has property A relative to a family of subgroups, each of which has property A, then the group has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with at least one stabilizer of property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.Comment: Updated to include a strengthening of the relative amenability characterizatio

Delineating the GRIN1 phenotypic spectrum: a distinct genetic NMDA receptor encephalopathy

Objective:To determine the phenotypic spectrum caused by mutations in GRIN1 encoding the NMDA receptor subunit GluN1 and to investigate their underlying functional pathophysiology.Methods:We collected molecular and clinical data from several diagnostic and research cohorts. Functional consequences of GRIN1 mutations were investigated in Xenopus laevis oocytes.Results:We identified heterozygous de novo GRIN1 mutations in 14 individuals and reviewed the phenotypes of all 9 previously reported patients. These 23 individuals presented with a distinct phenotype of profound developmental delay, severe intellectual disability with absent speech, muscular hypotonia, hyperkinetic movement disorder, oculogyric crises, cortical blindness, generalized cerebral atrophy, and epilepsy. Mutations cluster within transmembrane segments and result in loss of channel function of varying severity with a dominant-negative effect. In addition, we describe 2 homozygous GRIN1 mutations (1 missense, 1 truncation), each segregating with severe neurodevelopmental phenotypes in consanguineous families.Conclusions:De novo GRIN1 mutations are associated with severe intellectual disability with cortical visual impairment as well as oculomotor and movement disorders being discriminating phenotypic features. Loss of NMDA receptor function appears to be the underlying disease mechanism. The identification of both heterozygous and homozygous mutations blurs the borders of dominant and recessive inheritance of GRIN1-associated disorders.Johannes R. Lemke (32EP30_136042/1) and Peter De Jonghe (G.A.136.11.N and FWO/ESF-ECRP) received financial support within the EuroEPINOMICS-RES network (www.euroepinomics.org) within the Eurocores framework of the European Science Foundation (ESF). Saskia Biskup and Henrike Heyne received financial support from the German Federal Ministry for Education and Research (BMBF IonNeurONet: 01 GM1105A and FKZ: 01EO1501). Katia Hardies is a PhD fellow of the Institute for Science and Technology (IWT) Flanders. Ingo Helbig was supported by intramural funds of the University of Kiel, by a grant from the German Research Foundation (HE5415/3-1) within the EuroEPINOMICS framework of the European Science Foundation, and additional grants of the German Research Foundation (DFG, HE5415/5-1, HE 5415/6-1), German Ministry for Education and Research (01DH12033, MAR 10/012), and grant by the German chapter of the International League against Epilepsy (DGfE). The project also received infrastructural support through the Institute of Clinical Molecular Biology in Kiel, supported in part by DFG Cluster of Excellence "Inflammation at Interfaces" and "Future Ocean." The project was also supported by the popgen 2.0 network (P2N) through a grant from the German Ministry for Education and Research (01EY1103) and by the International Coordination Action (ICA) grant G0E8614N. Christel Depienne, Caroline Nava, and Delphine Heron received financial support for exome analyses by the Centre National de Genotypage (CNG, Evry, France)

A generalization of the Lyndon -Hochschild -Serre spectral sequence for polynomial cohomology

We construct an analogue of the Lyndon-Hochschild-Serre spectral sequence in the context of polynomial cohomology with E2-term H P*(Q; H P*(H; [special characters omitted])). For the polynomial extensions of Noskov, with the normal subgroup isocohomological, the spectral sequence converges to H P*( G; [special characters omitted]). In the case that both H and Q are isocohomological G must also be isocohomological. By referring to results of Connes-Moscovici and Noskov if H and Q are both isocohomological and have the Rapid Decay property, then G satisfies the Novikov conjecture