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1. Coarse models of homogeneous spaces and translations-like actions

   D. B. McReynolds, Mark Pengitore (January 2019, Preprint)

   For finitely generated groups G and H equipped with word metrics, a translation-like action of H on G is a free action where each element of H moves elements of G a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group G that is not isogenous to SL(2,ℝ) admit translation-like actions by ℤ2. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical N of the Borel subgroup AN of G acts translation-like on any cocompact lattice in G. We also prove that for noncompact simple Lie groups G,H with H 

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2. Dimension drop for diagonalizable flows on homogeneous spaces

   https://doi.org/10.3934/jmd.2024012  

   Kleinbock, Dmitry; Mirzadeh, Shahriar (January 2024, Journal of Modern Dynamics)

   Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms. 

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3. Zero–one laws for eventually always hitting points in rapidly mixing systems

   https://doi.org/10.4310/MRL.2023.v30.n3.a7  

   Kleinbock, Dmitry; Konstantoulas, Ioannis; Richter, Florian K (January 2023, Mathematical Research Letters)

   In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauß map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co-null set respectively). 

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4. Dynamical Borel–Cantelli lemma for recurrence under Lipschitz twists

   https://doi.org/10.1088/1361-6544/acafcb  

   Kleinbock, Dmitry; Zheng, Jiajie (January 2023, Nonlinearity)

   Abstract In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero–one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero–one laws for the two problems. 

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5. The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves

   https://doi.org/10.1090/ert/631  

   Frăţilă, Dragoş; Gunningham, Sam; Li, Penghui (December 2022, Representation Theory of the American Mathematical Society)

   We study the moduli stack of degree 0 0 semistable G G -bundles on an irreducible curve E E of arithmetic genus 1 1 , where G G is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H H of G G (the E E -pseudo-Levi subgroups), where each stratum is computed in terms of H H -bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g \mathfrak {g} (respectively, algebraic group G G ). We also provide a Tannakian description of these moduli stacks and use it to show that if E E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E E are equivariantly isomorphic to the unipotent cone in G G . Finally, we classify the E E -pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples. 

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