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1. Borel and Volume Classes for Dense Representations of Discrete Groups

   https://doi.org/10.1093/imrn/rnab078  

   Farre, James (April 2021, International Mathematics Research Notices)

   Abstract We show that the bounded Borel class of any dense representation $$\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $$G$$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $$\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$$ is uniformly separated in semi-norm from any other representation $$\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$$ for which there is a subgroup $$H\le G$$ on which $$\rho $$ is still dense but $$\rho ^{\prime}$$ is discrete or indiscrete but stabilizes a point, line, or plane in $${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of $${\operatorname{PSL}}_2{\mathbb{R}}$$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties. 

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2. Generalized versality, special points, and resolvent degree for the sporadic groups

   https://doi.org/10.1016/j.jalgebra.2024.02.025  

   Gómez-Gonzáles, Claudio; Sutherland, Alexander J; Wolfson, Jesse (June 2024, Journal of Algebra)

   Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group G has only been investigated when G is a cyclic group; an alternating group; a simple factor of a Weyl group of type E6, E7, or E8; or PSL(2,F7). In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) RDk≤d-versality, which we connect to the existence of “special points” on varieties. 

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3. Local-global principles in circle packings

   https://doi.org/10.1112/S0010437X19007139  

   Fuchs, Elena; Stange, Katherine E.; Zhang, Xin (June 2019, Compositio Mathematica)

   We generalize work by Bourgain and Kontorovich [ On the local-global conjecture for integral Apollonian gaskets , Invent. Math. 196 (2014), 589–650] and Zhang [ On the local-global principle for integral Apollonian 3-circle packings , J. Reine Angew. Math. 737 , (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$$ satisfying certain conditions, where $$K$$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that $${\mathcal{A}}$$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$$ containing a Zariski dense subgroup of $$\operatorname{PSL}_{2}(\mathbb{Z})$$ . 

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4. On the profinite rigidity of triangle groups

   M. R. Bridson, D. B. (January 2020, Preprint)

   We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups. 

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5. Minimal volume entropy of free-by-cyclic groups and 2-dimensional right-angled Artin groups

   https://doi.org/10.1007/s00208-021-02211-9  

   Bregman, Corey; Clay, Matt (April 2021, Mathematische Annalen)

   null (Ed.)

   Let G be a free-by-cyclic group or a 2-dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to G has minimal volume entropy equal to 0. In the nonvanishing case, we provide a positive lower bound to the minimal volume entropy of an aspherical simplicial complex of minimal dimension for these two classes of groups. Our results rely upon a criterion for the vanishing of the minimal volume entropy for 2-dimensional groups with uniform uniform exponential growth. This criterion is shown by analyzing the fiber π1-growth collapse and non-collapsing assumptions of Babenko-Sabourau. 

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