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Register a new userJoinings of higher rank torus actions on homogeneous spaces
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We show that joinings of higher rank torus actions on S-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
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Let $Gamma$ be a lattice in ${rm SL}(n, mathbb R)$ with $ngeq 3$ and $mathcal S$ be a closed surface. Then $Gamma$ has no distal minimal action on $mathcal S$.
We prove LeBrun--Salamon conjecture in the following situation: if $X$ is a contact Fano manifold of dimension $2n+1$ whose group of automorphisms is reductive of rank $geq max(2,(n-3)/2)$ then $X$ is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.
An important consequence of the theory of entropy of Z-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.
We evaluate the effective actions of supersymmetric matrix models on fuzzy S^2times S^2 up to the two loop level. Remarkably it turns out to be a consistent solution of IIB matrix model. Based on the power counting and SUSY cancellation arguments, we can identify the t Hooft coupling and large N scaling behavior of the effective actions to all orders. In the large N limit, the quantum corrections survive except in 2 dimensional limits. They are O(N) and O(N^{4over 3}) for 4 and 6 dimensional spaces respectively. We argue that quantum effects single out 4 dimensionality among fuzzy homogeneous spaces.
We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler implies that every action of a topological group $G$ on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Hellys selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.
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