Finiteness properties of cubulated groups

We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let H_1,...,H_s be relatively quasiconvex codimension-1 subgroups of a group G that is hyperbolic relative to P_1,...,P_r. We prove that G acts relatively cocompactly on the associated dual CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact when G is hyperbolic. When P_1,...,P_r are abelian, we show that the dual CAT(0) cube complex C has a G-cocompact CAT(0) truncation.Comment: 58 pages, 12 figures. Version 3: Revisions and slightly improved results in Sections 7 and 8. Several theorem numbers have changed from the previous versio

Isometric group actions on Banach spaces and representations vanishing at infinity

Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property ({\BP}_0^V), for $V$ be a Banach space: a locally compact group $G$ has property ({\BP}_0^V) if every affine isometric action of $G$ on $V$, such that the linear part is a $C_0$-representation of $G$, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have property ({\BP}_0^V). As a consequence for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on $L^2(G)$ is non-zero; and we characterize uniform lattices in those groups for which the first $L^2$-Betti number is non-zero.Comment: 28 page

Patients' satisfaction with community treatment: a pilot cross-sectional survey adopting multiple perspectives.

ACCESSIBLE SUMMARY: Patients' satisfaction is scarcely studied within the context of community treatment for adolescents. Thus, this study adopts a multiple perspective on patients' satisfaction (including service users as well as staff members). The results highlighted that all informants (patients, foster carers in foster homes and professional caregivers from community treatment teams) perceived the patients to be satisfied, with foster carers reporting the highest patient satisfaction rate. Considering the patient satisfaction rate from multiple perspectives provides complementary understandings. Clinical outcomes and, specifically, a reduction in emotional difficulties were related to patient's satisfaction, but only from the patients' perspective. ABSTRACT: Community treatment (CT) teams in Switzerland provide care to patients who are unable to use regular child and adolescent mental health services (i.e. inpatient and outpatients facilities). No study has considered patients' self-rated satisfaction alongside with staff members' perspectives on patient satisfaction. Thus, adopting a cross-sectional survey design, we collected patients' satisfaction using the Client Satisfaction Questionnaire (CSQ-8), rated by multiple informants (patients, foster carers in foster homes and professional caregivers from CT teams). Professional caregivers assessed clinical outcomes using the Health of the Nation Outcome Scale for Children and Adolescents. The results indicated that all informants were satisfied with the community treatment teams. The satisfaction scores were not correlated across informants; however, the alleviation of emotional symptoms was correlated with patients' satisfaction. This study indicated that the use of a combined approach including the views of service users and professionals gives important complementary information. Finally, in our sample, lower emotional symptoms were linked to enhanced patient satisfaction. This study demonstrated the importance of considering multiple perspectives to obtain the most accurate picture of patients' satisfaction. Second, focusing on the reduction of emotional symptoms might lead to a higher degree of patients' satisfaction

Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when 1. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

Towards ensemble asteroseismology of the young open clusters Chi Persei and NGC 6910

As a result of the variability survey in Chi Persei and NGC6910, the number of Beta Cep stars that are members of these two open clusters is increased to twenty stars, nine in NGC6910 and eleven in Chi Persei. We compare pulsational properties, in particular the frequency spectra, of Beta Cep stars in both clusters and explain the differences in terms of the global parameters of the clusters. We also indicate that the more complicated pattern of the variability among B type stars in Chi Persei is very likely caused by higher rotational velocities of stars in this cluster. We conclude that the sample of pulsating stars in the two open clusters constitutes a very good starting point for the ensemble asteroseismology of Beta Cep-type stars and maybe also for other B-type pulsators.Comment: 4 pages, Astronomische Nachrichten, HELAS IV Conference, Arecife, Lanzarote, Feb 2010, submitte

On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update