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1. Patterson–Sullivan measures for transverse subgroups

   https://doi.org/10.3934/jmd.2024009  

   Canary, Richard; Zhang, Tengren; Zimmer, Andrew (January 2024, Journal of Modern Dynamics)

   Forni, Giovanni (Ed.)

   We study Patterson–Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf–Tsuji–Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan Curve Theorem. We also use the Patterson–Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups. 

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2. TEMPERED HOMOGENEOUS SPACES IV

   https://doi.org/10.1017/s1474748022000287  

   Benoist, Yves; Kobayashi, Toshiyuki (June 2022, Journal of the Institute of Mathematics of Jussieu)

   Let 𝐺 be a complex semisimple Lie group and 𝘏 a complex closed connected subgroup. Let g and h be their Lie algebras. We prove that the regular representation of 𝐺 in 𝐿²(𝐺/𝘏) is tempered if and only if the orthogonal of h in g contains regular elements by showing simultaneously the equivalence to other striking conditions, such as h has a solvable limit algebra. 

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3. 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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4. Generating the homology of covers of surfaces

   https://doi.org/10.1112/blms.13026  

   Boggi, Marco; Putman, Andrew; Salter, Nick (March 2024, Bulletin of the London Mathematical Society)

   Abstract Putman and Wieland conjectured that if is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of under the action of lifts to of mapping classes on are infinite. We prove that this holds if is generated by the homology classes of lifts of simple closed curves on . We also prove that the subspace of spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2‐holed spheres, and we prove that is generated by the homology classes of lifts of loops on lying on subsurfaces homeomorphic to 3‐holed spheres. 

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5. Benjamini-Schramm convergence of periodic orbits

   https://doi.org/10.1007/s10711-022-00703-9  

   Mohammadi, Amir; Rafi, Kasra (August 2022, Geometriae Dedicata)

   Abstract We prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of translation surfaces. 

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