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

Consider a countable group Γ acting ergodically by measure preserving transformations on a probability space (X, µ), and let R Γ be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation R Γ on X determines the group Γ and the action (X, µ, Γ) uniquely, up to finite groups. The natural action of SL n (Z) on the n-torus R n /Z n , for n > 2, is one of such examples. The interpretation of these results in the context of von Neumann algebras provides some support to the conjecture of Connes on rigidity of group algebras for groups with property T. Our rigidity results also give examples of countable equivalence relations of type II 1 , which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.

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In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G.Example 1.2. Let Γ and Λ be lattices in the same locally compact second countable (lcsc) group G. Since G contains lattices it is necessarily unimodular, so its Haar measure m G is invariant under the left Γ-action γ : g → γ −1 g, and the right Λ-action λ : g → g λ, which obviously commute. Hence (G, m G ) forms a ME coupling for Γ and Λ.

Let ν \nu be a probability measure on S L d ( Z ) \mathrm {SL}_d(\mathbb {Z}) satisfying the moment condition E ν ( ‖ g ‖ ϵ ) > ∞ \mathbb {E}_\nu (\|g\|^\epsilon )>\infty for some ϵ \epsilon . We show that if the group generated by the support of ν \nu is large enough, in particular if this group is Zariski dense in S L d \mathrm {SL}_d , for any irrational x ∈ T d x \in \mathbb {T}^d the probability measures ν ∗ n ∗ δ x \nu ^{* n} * \delta _x tend to the uniform measure on T d \mathbb {T}^d . If in addition x x is Diophantine generic, we show this convergence is exponentially fast.

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We study rigidity properties of lattices in Isom(H n ) SO n,1 (R), n ≥ 3, and of surface groups in Isom(H 2 ) SL 2 (R) in the context of integrable measure equivalence. The results for lattices in Isom(H n ), n ≥ 3, are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n = 2 we show that cocompact lattices in Isom(H 2 ) allow a similar integrable measure equivalence classification.

A: PDE, partial diff erential equation.S S : V Z M Surface and subsurface fl ow systems are inherently unifi ed systems that are o en broken into sec ons for logical (e.g., me scales) and technical (e.g., analy cal and computa onal solvability) reasons. While the basic physical laws are common to surface and subsurface systems, spa al and temporal dimensions as well as the con nuum approach used for the subsurface lead to diff erent formula ons of the governing par al diff eren al equa ons. While in most applica ons such decoupling of the systems works well and allows a very accurate and effi cient descrip on of the individual system by trea ng the adjacent system as a boundary condi on, in the case of water fl ow over a porous medium, it does not. Therefore coupled models are in increasing use in this fi eld, led mostly by watershed and surface irriga on modelers.The governing equa ons of each component of the coupled system and the coupling physics and mathema cs are reviewed fi rst. Three diff erent coupling schemes are iden fi ed, namely the uncoupled (with the degenerated uncoupled scheme being a special case of the uncoupled), the itera vely coupled, and the fully coupled. Next, the diff erent applica ons of the diff erent coupling schemes, sorted by fi eld of applica on, are reviewed. Finally, some research gaps are discussed, led by the need to include ver cal momentum transfer and to expand the use of fully coupled models toward surface irriga on applica ons.F . 1. Rela ons between diff erent components of the hydrologic cycle. Black lines indicate vapor processes, blue lines indicate non-vapor processes. The solid, bold lines are the focus of this discussion.