Asymptotic Geometry in the product of Hadamard spaces with rank one isometries

In this article we study asymptotic properties of certain discrete groups $\Gamma$ acting by isometries on a product \XX=\XX_1\times \XX_2 of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $\Gamma$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $\delta_\theta(\Gamma)$ of orbit points in \XX with a prescribed so-called "slope" $\theta\in (0,\pi/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $\Psi_\Gamma$ to \RR_{\ge 0}^2 of $\delta_\theta(\Gamma)$ as a function of $\theta$ is upper semi-continuous and concave.Comment: 27 pages, to appear in Geometry & Topolog

Rotational Quantum Impurities in a Metal: Stability of the 2-Channel Kondo Fixed Point in a Magnetic Field

A three-level system with partially broken SU(3) symmetry immersed in a metal, comprised of a unique non-interacting ground state and two-fold degenerate excited states, exhibits a stable two-channel Kondo fixed point within a wide range of parameters, as has been shown in previous work. Such systems can, for instance, be realized by protons dissolved in a metal and bound in the interstitial space of the host lattice, where the degeneracy of excited rotational states is guaranteed by the space inversion symmetry of the lattice. We analyze the robustness of the 2CK fixed point with respect to a level splitting of the excited states and discuss how this may explain the behavior of the well-known dI/dV spectra measured by Ralph and Buhrman on ultrasmall quantum point contacts in a magnetic field.Comment: 7 pages, 3 figures; to appear in Ann. Physik (Berlin

Metric characterizations of spherical, and Euclidean buildings

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper17.abs.htm

On the analytic systole of Riemannian surfaces of finite type

In our previous work we introduced, for a Riemannian surface $S$, the quantity $\Lambda(S):=\inf_F\lambda_0(F)$, where $\lambda_0(F)$ denotes the first Dirichlet eigenvalue of $F$ and the infimum is taken over all compact subsurfaces $F$ of $S$ with smooth boundary and abelian fundamental group. A result of Brooks implies $\Lambda(S)\ge\lambda_0(\tilde{S})$, the bottom of the spectrum of the universal cover $\tilde{S}$. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate $\Lambda(S)$ with the systole, improving a result by the last named author.Comment: 35 pages, 1 figure; v2: slightly reorganized, fixed a technical problem in the proof of Thm. 7.3 (v2), added some references, to appear in GAF

Eigenvalues and Holonomy

We estimate the eigenvalues of connection Laplacians in terms of the non-triviality of the holonomy.Comment: 9 page

On the bottom of spectra under coverings

For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with boundary (possibly empty) and respective fundamental groups $\Gamma_1\subseteq\Gamma_0$, we show that the bottoms of the spectra of $M_0$ and $M_1$ coincide if the right action of $\Gamma_0$ on $\Gamma_1\backslash\Gamma_0$ is amenable.Comment: 8 pages, fixed a technical mistake concerning the volume of the boundary of fundamental domain

Small eigenvalues of surfaces - old and new

We discuss our recent work on small eigenvalues of surfaces. As an introduction, we present and extend some of the by now classical work of Buser and Randol and explain novel ideas from articles of S\'evennec, Otal, and Otal-Rosas which are of importance in our line of thought.Comment: 24 pages, 5 figures, all comments welcom

Boundary Value Problems for Elliptic Differential Operators of First Order

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.Comment: 79 pages, 6 figures, minor corrections, references adde