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1. Actions of 𝐴𝑙𝑡(𝑛) on groups of finite Morley rank without involutions

   https://doi.org/10.1090/proc/16530  

   Altınel, Tuna; Wiscons, Joshua (January 2024, Proceedings of the American Mathematical Society)

   We investigate faithful representations of Alt(n) as automorphisms of a connected group 𝐺 of finite Morley rank. We target a lower bound of 𝑛 on the rank of such a nonsolvable 𝐺, and our main result achieves this in the case when 𝐺 is without involutions. In the course of our analysis, we also prove a corresponding bound for solvable 𝐺 by leveraging recent results on the abelian case. We conclude with an application towards establishing natural limits to the degree of generic transitivity for permutation groups of finite Morley rank. 

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2. GROUPS OF MORLEY RANK 4

   https://doi.org/10.1017/jsl.2014.81  

   WISCONS, JOSHUA (March 2016, The Journal of Symbolic Logic)

   Abstract We show that any simple group of Morley rank 4 must be a bad group with no proper definable subgroups of rank larger than 1. We also give an application to groups acting on sets of Morley rank 2. 

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3. TRANSITIVITY, LOWNESS, AND RANKS IN NSOP THEORIES

   https://doi.org/10.1017/jsl.2023.36  

   CHERNIKOV, ARTEM; KIM, BYUNGHAN; RAMSEY, NICHOLAS (June 2023, The Journal of Symbolic Logic)

   Abstract We develop the theory of Kim-independence in the context of NSOP $$_{1}$$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $$_{1}$$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $$_{1}$$ theories. 

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4. Positively curved Riemannian metrics with logarithmic symmetry rank bounds

   https://doi.org/10.4171/CMH/340  

   Kennard, Lee (November 2014, Commentarii Mathematici Helvetici)

   We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group. 

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5. Invariant Measures for Horospherical Actions and Anosov Groups

   https://doi.org/10.1093/imrn/rnac262  

   Lee, Minju; Oh, Hee (October 2022, International Mathematics Research Notices)

   Abstract Let $$\Gamma $$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $$G$$. For a maximal horospherical subgroup $$N$$ of $$G$$, we show that the space of all non-trivial $NM$-invariant ergodic and $$A$$-quasi-invariant Radon measures on $$\Gamma \backslash G$$, up to proportionality, is homeomorphic to $${\mathbb {R}}^{\text {rank}\,G-1}$$, where $$A$$ is a maximal real split torus and $$M$$ is a maximal compact subgroup that normalizes $$N$$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures. 

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