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1. On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics

   https://doi.org/10.1093/imrn/rnab120  

   Ebenfelt, Peter; Xiao, Ming; Xu, Hang (June 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the Bergman metric of a finite ball quotient $$\mathbb{B}^n/\Gamma $$, where $$n \geq 2$$ and $$\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $$\Gamma $$ is trivial, that is, when the ball quotient $$\mathbb{B}^n/\Gamma $$ is the unit ball $${\mathbb{B}}^n$$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric. 

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2. Equivariant hyperbolization of 3-manifolds via homology cobordisms

   Dave Auckly, Hee Jung (January 2023, Topology and its applications)

   The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Zp-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order. 

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3. The spread of a finite group

   https://doi.org/doi.org/10.4007/annals.2021.193.2.5  

   Burness, Tim; Guralnick, Robert; Harper, Scott (January 2021, Annals of mathematics)

   null (Ed.)

   A group G is said to be 3/2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property, then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ϵ G, there exists y ϵ G such that G = ⟨x1, y⟩ = ⟨x2, y⟩. In other words, s(G) ⩾ 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods, and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type, and this is the case we treat in this paper. 

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4. Largest hyperbolic action of 3‐manifold groups

   https://doi.org/10.1112/blms.13118  

   Abbott, Carolyn; Nguyen, Hoang Thanh; Rasmussen, Alexander J (October 2024, Bulletin of the London Mathematical Society)

   Abstract The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. Following Abbott–Balasubramanya–Osin, the group is ‐accessibleif the resulting poset has a largest element. In this paper, we prove that every nongeometric 3‐manifold has a finite cover with ‐inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is ‐inaccessible. We also prove that every Croke–Kleiner admissible group (a class of graphs of groups that generalizes fundamental groups of three‐dimensional graph manifolds) has a finite index subgroup that is ‐inaccessible. 

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5. CR embeddability of quotients of the Rossi sphere via spectral theory

   https://doi.org/10.1142/S0129167X22500148  

   Bosch, Henry; Gonzales, Tyler; Spinelli, Kamryn; Udell, Gabe; Zeytuncu, Yunus E. (February 2022, International Journal of Mathematics)

   We look at the action of finite subgroups of [Formula: see text] on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of [Formula: see text] to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum. 

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