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1. THE NUMBER OF SET ORBITS OF PERMUTATION GROUPS AND THE GROUP ORDER

   https://doi.org/10.1017/S0004972721001064  

   GINTZ, MICHAEL; KELLER, THOMAS M.; KORTJE, MATTHEW; WANG, ZILI; YANG, YONG (August 2022, Bulletin of the Australian Mathematical Society)

   Abstract If G is permutation group acting on a finite set $$\Omega $$ , then this action induces a natural action of G on the power set $$\mathscr{P}(\Omega )$$ . The number $s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, $$\inf ({\log _2 s(G)}/{\log _2 |G|})$$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any $${{\textrm {A}}}_l, l> 4$$ , as a composition factor. 

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2. On asymptotic stability of connective groups

   https://doi.org/doi:10.2969/aspm/08010000  

   Dadarlat, Marius (April 2019, Operator algebras and mathematical physics)

   In their study of almost group representations, Manuilov and Mishchenko introduced and investigated the notion of asymptotic stability of a finitely presented discrete group. In this paper we establish connections between connectivity of amenable groups and asymptotic stability and exhibit new classes of asymptotically stable groups. In particular, we show that if G is an amenable and connective discrete group whose classifying space BG is homotopic to a finite simplicial complex, then G is asymptotically stable. 

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3. 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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4. 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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5. Quantum representations and monodromies of fibered links.

   Renaud Detcherry, Efstratia Kalfagianni (July 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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