An intermediate quasi-isometric invariant between subexponential asymptotic dimension growth and Yu's Property A

We present the notion of asymptotically large depth for a metric space which is (a priory) weaker than having subexponential asymptotic dimension growth and (a priory) stronger than property A.Comment: 13 pages, some typos were correcte

Fixed point theorem for reflexive Banach spaces and uniformly convex non positively curved metric spaces

This article generalizes the work of Ballmann and \'Swiatkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes

High dimensional analogue of metric distortion for simplicial complexes

We suggest a new possible high dimensional analogue to metric distortion. We then show a possible method for providing lower bounds to this distortion and use this method to prove a "Bourgain-type" distortion theorem for Linial-Meshulam random complexes.Comment: 21 page

Isoperimetric Inequalities and topological overlapping for quotients of Affine buildings

We prove isoperimetric inequalities for quotients of $n$-dimensional Affine buildings. We use these inequalities to prove topological overlapping for the 2-dimensional skeletons of these buildings.Comment: 40 pages. Replacement of previous version which referred only to isoperimetric inequalities for 1-cochains (and not to topological overlapping

Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps

This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.Comment: As advised by the referees, the arxiv paper "Local spectral expansion approach to high dimensional expanders" arXiv:1407.8517 was split into two parts since it was too long to publish as a whole. This is part I of the split articl

Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical overlapping

In this paper, we further explore the local-to-global approach for expansion of simplicial complexes that we call local spectral expansion. Specifically, we prove that local expansion in the links imply the global expansion phenomena of mixing and geometric overlapping. Our mixing results also give tighter bounds on the error terms compared to previously known results.Comment: 31 page

Local spectral expansion approach to high dimensional expanders

This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We show the condition of local spectral expansion has several nice implications. For example, for a simplicial complex with local spectral expansion we show vanishing of cohomology with real coefficients, Cheeger type inequalities and mixing type results and geometric overlap results.Comment: 96 pages. This version has minor corrections regarding the result of mixing for partite complexe

Angle criteria for uniform convergence of averaged projections and cyclic or random products of projections

We apply a new notion of angle between projections to deduce criteria for uniform convergence results of the alternating projections method under several different settings: averaged projections, cyclic products, quasi-periodic products and random products.Comment: 13 pages, some typos were correcte

Averaged projections, angles between groups and strengthening of property (T)

Using the method of averaged projections and introducing a new notion of angles between projections, we establish a criterion for a certain type of strengthening of property (T) (which is weaker than the notion of strong Banach property (T) introduced by Lafforgue). We also derive several applications regarding fixed point properties and Banach expanders and give examples of these applications.Comment: 39 pages, added results regarding the connection between angle and graph Laplacian and improved results regarding groups acting on simplicial complexe

Strong geodesic convex functions of order m

Strong geodesic convex function and strong monotone vector field of order $m$ on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order $m$ for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequality problem and the strict minimizers of order $m$ for a multiobjective programming problem has also been established.Comment: 10 pages, No figur