More Like this

1. 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. 

   more » « less
2. Absolute profinite rigidity and hyperbolic geometry

   M. R. Bridson, D. B. (January 2020, Annals of mathematics)

   We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group PSL(2,ℤ[ω]) with ω2+ω+1=0 is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in PSL(2,ℂ) and the fundamental group of the Weeks manifold (the closed hyperbolic 3-manifold of minimal volume). 

   more » « less
3. Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

   https://doi.org/10.1353/ajm.2024.a928325  

   Kuhn, Nicholas J; Lloyd, Christopher_J R (June 2024, American Journal of Mathematics)

   abstract: In the early 1940s, P. A. Smith showed that if a finite $$p$$-group $$G$$ acts on a finite dimensional complex $$X$$ that is mod $$p$$ acyclic, then its space of fixed points, $X^G$, will also be mod $$p$$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $$G$$-space $$X$$ is a equivariant homotopy retract of the $$p$$-localization of a based finite $$G$$-C.W. complex, given $H 

   more » « less
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. 

   more » « less
5. On Polish groups admitting non-essentially countable actions

   https://doi.org/10.1017/etds.2020.133  

   KECHRIS, ALEXANDER S.; MALICKI, MACIEJ; PANAGIOTOPOULOS, ARISTOTELIS; ZIELINSKI, JOSEPH (January 2022, Ergodic Theory and Dynamical Systems)

   Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable. 

   more » « less