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1. SIMPLE GROUPS OF MORLEY RANK 5 ARE BAD

   https://doi.org/10.1017/jsl.2017.86  

   DELORO, ADRIEN; WISCONS, JOSHUA (September 2018, The Journal of Symbolic Logic)

   Abstract We show that any simple group of Morley rank 5 is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most 2. The main result is then used to catalog the nonsoluble connected groups of Morley rank 5. 

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2. 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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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. ASSOCIATIVITY OF THE MORLEY PRODUCT OF INVARIANT MEASURES IN NIP THEORIES

   https://doi.org/10.1017/jsl.2021.55  

   CONANT, GABRIEL; GANNON, KYLE (September 2021, The Journal of Symbolic Logic)

   Abstract In light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories. 

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5. Rank of Matrices with Entries from a Multiplicative Group

   https://doi.org/10.1093/imrn/rnac183  

   Alon, Noga; Solymosi, József (July 2022, International Mathematics Research Notices)

   Abstract We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $$\mathbb {R}^d$$ generate high-rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets. 

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