More Like this

1. Stability for hyperbolic groups acting on boundary spheres

   https://doi.org/10.1017/fms.2023.78  

   Mann, Kathryn; Manning, Jason Fox (January 2023, Forum of Mathematics, Sigma)

   Abstract A hyperbolic groupGacts by homeomorphisms on its Gromov boundary. We show that if$$\partial G$$is a topologicaln–sphere, the action istopologically stablein the dynamical sense: any nearby action is semi-conjugate to the standard boundary action. 

   more » « less
2. Bohr sets in sumsets II: countable abelian groups

   https://doi.org/10.1017/fms.2023.49  

   Griesmer, John T.; Le, Anh N.; Lê, Thái Hoàng (July 2023, Forum of Mathematics, Sigma)

   Abstract We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, lettingGbe a countable discrete abelian group and$$\phi _1, \phi _2, \phi _3: G \to G$$be commuting endomorphisms whose images have finite indices, we show that(1)If$$A \subset G$$has positive upper Banach density and$$\phi _1 + \phi _2 + \phi _3 = 0$$, then$$\phi _1(A) + \phi _2(A) + \phi _3(A)$$contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$$\mathbb {Z}$$and a recent result of the first author.(2)For any partition$$G = \bigcup _{i=1}^r A_i$$, there exists an$$i \in \{1, \ldots , r\}$$such that$$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$$contains a Bohr set. This generalizes a result of the second and third authors from$$\mathbb {Z}$$to countable abelian groups.(3)If$$B, C \subset G$$have positive upper Banach density and$$G = \bigcup _{i=1}^r A_i$$is a partition,$$B + C + A_i$$contains a Bohr set for some$$i \in \{1, \ldots , r\}$$. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices$$[G:\phi _j(G)]$$, the upper Banach density ofA(in (1)), or the number of sets in the given partition (in (2) and (3)). 

   more » « less
3. Local parameters of supercuspidal representations

   https://doi.org/10.1017/fmp.2024.10  

   Gan, Wee Teck; Harris, Michael; Sawin, Will; Beuzart-Plessis, Raphaël (January 2024, Forum of Mathematics, Pi)

   Abstract For a connected reductive groupGover a nonarchimedean local fieldFof positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter$${\mathcal {L}}^{ss}(\pi )$$to each irreducible representation$$\pi $$. Our first result shows that the Genestier-Lafforgue parameter of a tempered$$\pi $$can be uniquely refined to a tempered L-parameter$${\mathcal {L}}(\pi )$$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of$${\mathcal {L}}^{ss}(\pi )$$for unramifiedGand supercuspidal$$\pi $$constructed by induction from an open compact (modulo center) subgroup. If$${\mathcal {L}}^{ss}(\pi )$$is pure in an appropriate sense, we show that$${\mathcal {L}}^{ss}(\pi )$$is ramified (unlessGis a torus). If the inducing subgroup is sufficiently small in a precise sense, we show$$\mathcal {L}^{ss}(\pi )$$is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is$${\mathbb {P}}^1$$and a simple application of Deligne’s Weil II. 

   more » « less
4. $L^p$ regularity of the Bergman projection on the symmetrized polydisc

   https://doi.org/10.4153/S0008414X24000634  

   Huo, Zhenghui; Wick, Brett D (October 2024, Canadian Journal of Mathematics)

   Abstract We study the$$L^p$$regularity of the Bergman projectionPover the symmetrized polydisc in$$\mathbb C^n$$. We give a decomposition of the Bergman projection on the polydisc and obtain an operator equivalent to the Bergman projection over antisymmetric function spaces. Using it, we obtain the$$L^p$$irregularity ofPfor$$p=\frac {2n}{n-1}$$which also implies thatPis$$L^p$$bounded if and only if$$p\in (\frac {2n}{n+1},\frac {2n}{n-1})$$. 

   more » « less
5. 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. 

   more » « less