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1. Cusps and commensurability classes ofhyperbolic 4–manifolds

   https://doi.org/10.2140/agt.2023.23.3805  

   Sell, Connor (January 2023, Algebraic & Geometric Topology)

   There are six orientable compact flat 3–manifolds that can occur as cusp cross-sections of hyperbolic 4–manifolds. We provide criteria for exactly when a given commensurability class of arithmetic hyperbolic 4–manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type. 

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2. Gromov norm and Turaev-Viro invariants of 3-manifolds

   Detcherry, Renaud; Kalfagianni, Efstratia (December 2020, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $$TV_r $$and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $$M$$, with (possibly empty) toroidal boundary, $log|TVr(M)|$ is bounded above by a function linear in $$r$$ and whose slope is a positive universal constant times the Gromov norm of $$M$$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $$log|TVr(M)|\geq B r$$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$- Turaev-Viro invariants grow exponentially. These constructions are essential for the results of article [3] where we make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants TVr and hyperbolic volume are preserved under Dehn filling. Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm. 

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3. Reflections on trisection genus

   Chu, Michelle; Tillmann, Stephan (January 2019, Revue roumaine de mathématiques pures et appliquées)

   Fukui, Toshizumi; Koike, Satoshi; Paunescu, Laurențiu (Ed.)

   The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that analogous results concerning the trisection genus of a smooth, orientable 4-manifold have more general answers than their counterparts for 3-manifolds. In the case of hyperbolic 4-manifolds, upper and lower bounds are given in terms of volume and a trisection of the Davis manifold is described. 

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4. Convex co‐compact representations of 3‐manifold groups

   https://doi.org/10.1112/topo.12332  

   Islam, Mitul; Zimmer, Andrew (May 2024, Journal of Topology)

   Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples. 

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5. Extensions of multicurve stabilizers are hierarchically hyperbolic

   Russell, Jacob (December 2024, ArXiv)

   For a closed and orientable surface of genus at least 2, we prove the surface group extensions of the stabilizers of multicurves are hierarchically hyperbolic groups. This answers a question of Durham, Dowdall, Leininger, and Sisto. We also include an appendix that employs work of Charney, Cordes, and Sisto to characterize the Morse boundaries of hierarchically hyperbolic groups whose largest acylindrical action on a hyperbolic space is on a quasi-tree. 

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