Tessellations of homogeneous spaces of classical groups of real rank two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples. It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, R.Kulkarni and T.Kobayashi constructed examples that are not obvious when G = SO(2,2n) or SU(2,2n). H.Oh and D.Witte constructed additional examples in both of these cases, and obtained a complete classification when G = SO(2,2n). We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G = SU(2,2n). This includes the construction of another family of examples. The main results are obtained from methods of Y.Benoist and T.Kobayashi: we fix a Cartan decomposition G = KAK, and study the intersection of KHK with A. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.Comment: 74 pages, 7 figure

Integrality of Volumes of Representations

Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [BucherBurgerIozzi2013] we show that the volume of a representation of the fundamental group of M into the connected component of the isometry group of hyperbolic n-space, properly normalized, takes integer values if n=2m is at least 4. If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.Comment: According to the suggestions of the referee, the article has been almost completely rewritten with the respect to the first versio

Higher Teichm\"uller Spaces: from SL(2,R) to other Lie groups

The first part of this paper surveys several characterizations of Teichm\"uller space as a subset of the space of representation of the fundamental group of a surface into PSL(2,R). Special emphasis is put on (bounded) cohomological invariants which generalize when PSL(2,R) is replaced by a Lie group of Hermitian type. The second part discusses underlying structures of the two families of higher Teichm\"uller spaces, namely the space of maximal representations for Lie groups of Hermitian type and the space of Hitchin representations or positive representations for split real simple Lie groups.Comment: The file uploaded on May 12th was the wrong one and did not contain the Section 4.6 that was added. This is the version to appear in the Handbook of Teichm\"uller theor

Short Term and Long Term Effects of Price Cap Regulation

This paper uses a very simple example (two goods, linear symmetric demand and cost) to study the effects of the price cap regulatory mechanism. We show that if a given price vector is preferred (using current welfare as the criterion) to another, then it is not necessarily the case that it is also preferred in the long run (using the presented discounted value of welfare as the criterion). The relationship between current welfare and profit and therefore the firm's incentive to bargain for a given price vector depend on the specific details of the mechanism considered.Ramsey prices; Price cap regulation.

The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes

We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients --of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the Appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof we construct a \Gamma-equivariant measurable map from a Poisson boundary of \Gamma with values in the non-terminating ultrafilters on the Roller boundary of X.Comment: Minor changes that clarify some confusion have been made. Some figures have been adde