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1. Punctured groups for exotic fusion systems

   https://doi.org/10.1112/tlm3.12054  

   Henke, Ellen; Libman, Assaf; Lynd, Justin (December 2023, Transactions of the London Mathematical Society)

   Abstract The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the ‐local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are, however, not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups , a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite ‐group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson–Solomon exotic fusion system at the prime 2 has a punctured group, while the others do not. As for exotic fusion systems at odd primes , we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial ‐groups of order . 

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2. COUNTING CONJUGACY CLASSES IN

   https://doi.org/10.1017/S0004972718000047  

   HULL, MICHAEL; KAPOVICH, ILYA (June 2018, Bulletin of the Australian Mathematical Society)

   We show that if a finitely generated group $$G$$ has a nonelementary WPD action on a hyperbolic metric space $$X$$ , then the number of $$G$$ -conjugacy classes of $$X$$ -loxodromic elements of $$G$$ coming from a ball of radius $$R$$ in the Cayley graph of $$G$$ grows exponentially in $$R$$ . As an application we prove that for $$N\geq 3$$ the number of distinct $$\text{Out}(F_{N})$$ -conjugacy classes of fully irreducible elements $$\unicode[STIX]{x1D719}$$ from an $$R$$ -ball in the Cayley graph of $$\text{Out}(F_{N})$$ with $$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$$ of the order of $$R$$ grows exponentially in $$R$$ . 

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3. Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras

   https://doi.org/10.1093/imrn/rnz170  

   Xie, Zhizhang; Yu, Guoliang (August 2019, International Mathematics Research Notices)

   Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $$K_0$$-group of its group $$C^\ast$$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group. 

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4. ON ALPERIN’S LOWER BOUND FOR THE NUMBER OF BRAUER CHARACTERS

   https://doi.org/10.1007/S00031-022-09734-8  

   MALLE, GUNTER; NAVARRO, GABRIEL; TIEP, PHAM HUU (September 2023, Transformation Groups)

   Abstract We prove that the number of conjugacy classes of a finite groupGconsisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup ofG. This is predicted by the Alperin Weight Conjecture. 

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5. Counting conjugacy classes of fully irreducibles: double exponential growth

   https://doi.org/10.1007/s10711-024-00885-4  

   Kapovich, Ilya; Pfaff, Catherine (April 2024, Geometriae Dedicata)

   Inspired by results of Eskin and Mirzakhani (J Mod Dyn 5(1):71–105, 2011) counting closed geodesics of length at most L in the moduli space of a fixed closed surface, we consider a similar question in the Out(Fr) setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilatations have natural logarithm at most L. Let N(L) denote the number of Out(Fr)-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is at most L. We prove for r>2 that as L goes to infinity, the number N(L) has double exponential (in L) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems. 

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