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1. COMPATIBLE SYSTEMS OF GALOIS REPRESENTATIONS ASSOCIATED TO THE EXCEPTIONAL GROUP

   https://doi.org/10.1017/fms.2018.24  

   BOXER, GEORGE; CALEGARI, FRANK; EMERTON, MATTHEW; LEVIN, BRANDON; MADAPUSI PERA, KEERTHI; PATRIKIS, STEFAN (January 2019, Forum of Mathematics, Sigma)

   We construct, over any CM field, compatible systems of $$l$$ -adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $$l$$ ) algebraic monodromy groups equal to the exceptional group of type $$E_{6}$$ . 

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2. Pseudo-Anosov subgroups of general fibered 3–manifold groups

   https://doi.org/10.1090/btran/157  

   Leininger, Christopher; Russell, Jacob (August 2023, Transactions of the American Mathematical Society, Series B)

   We show that finitely generated and purely pseudo-Anosov subgroups of fundamental groups of fibered 3–manifolds with reducible monodromy are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. Combined with results of Dowdall–Kent–Leininger and Kent–Leininger–Schleimer, this establishes the result for the image of all such fibered 3–manifold groups in the mapping class group. 

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3. An arithmetic Kontsevich–Zorich monodromy of a symmetric origami in genus 4

   Gong, Xun; Sanchez, Anthony (December 2023, New York journal of mathematics)

   We demonstrate the existence of a certain genus four origami whose Kontsevich--Zorich monodromy is arithmetic in the sense of Sarnak. The surface is interesting because its Veech group is as large as possible and given by SL(2,Z). When compared to other surfaces with Veech group SL(2,Z) such as the Eierlegendre Wollmichsau and the Ornithorynque, an arithmetic Kontsevich--Zorich monodromy is surprising and indicates that there is little relationship between the Veech group and monodromy group of origamis. Additionally, we record the index and congruence level in the ambient symplectic group which gives data on what can appear in genus 4. 

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4. Definable structures on flat bundles

   https://doi.org/10.1112/blms.12879  

   Bakker, Benjamin; Mullane, Scott (July 2023, Bulletin of the London Mathematical Society)

   Abstract A flat vector bundle on an algebraic variety supports two natural definable structures given by the flat and algebraic coordinates. In this note, we show these two structures are compatible, subject to a condition on the local monodromy at infinity that is satisfied for all flat bundles underlying variations of Hodge structures. 

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5. Monodromy of Kodaira fibrations of genus 3

   https://doi.org/10.1002/mana.202000189  

   Flapan, Laure (November 2022, Mathematische Nachrichten)

   Abstract A Kodaira fibration is a non‐isotrivial fibration from a smooth algebraic surfaceSto a smooth algebraic curveBsuch that all fibers are smooth algebraic curves of genusg. Such fibrations arise as complete curves inside the moduli space of genusgalgebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of parametrizing curves whose Jacobians have extra endomorphisms. 

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