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1. Shapes of hyperbolic triangles and once-punctured torus groups

   https://doi.org/10.1007/s00209-021-02745-3  

   Kim, Sang-hyun; Koberda, Thomas; Lee, Jaejeong; Ohshika, Ken’ichi; Tan, Ser Peow; Gao, Xinghua (January 2021, Mathematische Zeitschrift)

   null (Ed.)

   Abstract Let $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds. 

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2. Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

   https://doi.org/10.1093/imrn/rnae126  

   Ramadas, Rohini (June 2024, International Mathematics Research Notices)

   Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $$n$$th Gleason polynomial $$G_{n}\in{\mathbb{Q}}[c]$$ comprise the $$0$$-dimensional moduli space of quadratic polynomials with an $$n$$-periodic critical point. $$\operatorname{Per}_{n}(0)$$ is the $$1$$-dimensional moduli space of quadratic rational maps on $${\mathbb{P}}^{1}$$ with an $$n$$-periodic critical point. We show that if $$G_{n}$$ is irreducible over $${\mathbb{Q}}$$, then $$\operatorname{Per}_{n}(0)$$ is irreducible over $${\mathbb{C}}$$. To do this, we exhibit a $${\mathbb{Q}}$$-rational smooth point on a projective completion of $$\operatorname{Per}_{n}(0)$$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $$n$$, $$\operatorname{Per}_{n}(0)$$ itself has no $${\mathbb{Q}}$$-rational points. 

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3. Rationality of twists of the Siegel modular variety of genus 2 and level 3

   https://doi.org/10.1090/proc/15854  

   Calegari, Frank; Chidambaram, Shiva (February 2022, Proceedings of the American Mathematical Society)

   Let  ρ ¯ : G Q → GSp 4 ⁡ ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the  3 3 -torsion of a principally polarized abelian surface  A / Q A/\mathbf {Q} . We prove that the moduli space  A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces  B / Q B/\mathbf {Q} admitting a symplectic isomorphism  B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over  Q \mathbf {Q} when  ρ ¯ \overline {\rho } is surjective, even though it is both rational over  C \mathbf {C} and unirational over  Q \mathbf {Q} via a map of degree  6 6 . 

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4. Character Ratios for Exceptional Groups of Lie Type

   https://doi.org/10.1093/imrn/rnz206  

   Liebeck, Martin W.; Tiep, Pham Huu (October 2019, International Mathematics Research Notices)

   Abstract We prove character ratio bounds for finite exceptional groups $G(q)$ of Lie type. These take the form $$\dfrac{|\chi (g)|}{\chi (1)} \le \dfrac{c}{q^k}$$ for all nontrivial irreducible characters $$\chi$$ and nonidentity elements $$g$$, where $$c$$ is an absolute constant, and $$k$$ is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs. 

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5. The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems

   https://doi.org/10.1090/bproc/203  

   Belmont, Eva; Xu, Zhouli; Zhang, Shangjie (January 2024, Proceedings of the American Mathematical Society, Series B)

   We describe in terms of generators and relations the ring structure of the $RO\left(C_2\right)$-graded $C_2$-equivariant stable stems ${\mathrm{\pi <\#comment/>}}_{\star <\#comment/>}^{C_2}$modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational $C_2$-equivariant sphere ${\mathrm{\pi <\#comment/>}}_{\star <\#comment/>}^{C_2}\left({\mathbb{S}}_{\mathbb{Q}}\right)$. 

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