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1. Fixed point sets and the fundamental group I: semi-free actions on G -CW-complexes

   https://doi.org/10.1017/prm.2023.63  

   Cappell, Sylvain; Weinberger, Shmuel; Yan, Min (August 2023, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Smith theory says that the fixed point set of a semi-free action of a group$$G$$on a contractible space is$${\mathbb {Z}}_p$$-acyclic for any prime factor$$p$$of the order of$$G$$. Jones proved the converse of Smith theory for the case$$G$$is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain$$K$$-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of$$K$$-theoretical obstructions. 

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2. On the weight zero compactly supported cohomology of H_{g,n}

   https://doi.org/10.1017/fms.2024.53  

   Brandt, Madeline; Chan, Melody; Kannan, Siddarth (January 2024, Forum of Mathematics, Sigma)

   Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves. 

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3. G -valued crystalline deformation rings in the Fontaine–Laffaille range

   https://doi.org/10.1112/S0010437X23007297  

   Booher, Jeremy; Levin, Brandon (August 2023, Compositio Mathematica)

   Let$$G$$be a split reductive group over the ring of integers in a$$p$$-adic field with residue field$$\mathbf {F}$$. Fix a representation$$\overline {\rho }$$of the absolute Galois group of an unramified extension of$$\mathbf {Q}_p$$, valued in$$G(\mathbf {F})$$. We study the crystalline deformation ring for$$\overline {\rho }$$with a fixed$$p$$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$$G$$-valued representations. In particular, we give a root theoretic condition on the$$p$$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. 

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4. Khintchine-type double recurrence in abelian groups

   https://doi.org/10.1017/etds.2024.29  

   ACKELSBERG, ETHAN (April 2024, Ergodic Theory and Dynamical Systems)

   Abstract We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if$$\Gamma $$is a countable discrete abelian group,$$\varphi , \psi \in \mathrm {End}(\Gamma )$$, and$$\psi - \varphi $$is an injective endomorphism with finite index image, then for any ergodic measure-preserving$$\Gamma $$-system$$( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$$, any measurable set$$A \in {\mathcal {X}}$$, and any$${\varepsilon }> 0$$, there is a syndetic set of$$g \in \Gamma$$such that$$\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$$. This generalizes the main results of Ackelsberget al[Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107]. For the group$$\Gamma = {\mathbb {Z}}^d$$, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to thequasi-affine(orConze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to$$\varphi $$and$$\psi $$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. 

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5. Cyclic base change of cuspidal automorphic representations over function fields

   https://doi.org/10.1112/S0010437X24007243  

   Böckle, Gebhard; Feng, Tony; Harris, Michael; Khare, Chandrashekhar B; Thorne, Jack A (September 2024, Compositio Mathematica)

   Let$$G$$be a split semisimple group over a global function field$$K$$. Given a cuspidal automorphic representation$$\Pi$$of$$G$$satisfying a technical hypothesis, we prove that for almost all primes$$\ell$$, there is a cyclic base change lifting of$$\Pi$$along any$$\mathbb {Z}/\ell \mathbb {Z}$$-extension of$$K$$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group$$G$$over a local function field$$F$$, and almost all primes$$\ell$$, any irreducible admissible representation of$$G(F)$$admits a base change along any$$\mathbb {Z}/\ell \mathbb {Z}$$-extension of$$F$$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi. 

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