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1. Virtually special embeddings of integral Lorentzian lattices

   https://doi.org/10.1090/conm/760/15285  

   Chu, Michelle (January 2020, Contemporary mathematics)

   Collin, Olivier; Friedl, Stefan; Gordon, Cameron; Tillmann, Stephan; Watson, Liam (Ed.)

   The automorphism groups of integral Lorentzian lattices act by isometries on hyperbolic space with finite covolume. In the case of reflective integral lattices, the automorphism groups are commensurable to arithmetic hyperbolic reflection groups. However, for a fixed dimension, there is only finitely many reflective integral Lorentzian lattices, and these can only occur in small dimensions. The goal of this note is to construct embeddings of lowdimensional integral Lorentzian lattices into unimodular Lorentzian lattices associated to right-angled reflection groups. As an application, we construct many discrete groups of Isom(Hn) for small n which are C-special in the sense of Haglund-Wise. 

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2. PLATEAU PROBLEMS FOR MAXIMAL SURFACES IN PSEUDO-HYPERBOLIC SPACE

   Labourie, Francois; Toulisse, Jeremy; Wolf, Michael (January 2024, Annales Scientifiques de lEcole Normale Supérieure)

   We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the pseudo-hyperbolic space which are limits of positive curves. We also discuss a compact Plateau problem. The required compactness arguments rely on an analysis of the pseudo-holomorphic curves defined by the Gauß lifts of the maximal surfaces. 

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3. Invariant Measures for Horospherical Actions and Anosov Groups

   https://doi.org/10.1093/imrn/rnac262  

   Lee, Minju; Oh, Hee (October 2022, International Mathematics Research Notices)

   Abstract Let $$\Gamma $$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $$G$$. For a maximal horospherical subgroup $$N$$ of $$G$$, we show that the space of all non-trivial $NM$-invariant ergodic and $$A$$-quasi-invariant Radon measures on $$\Gamma \backslash G$$, up to proportionality, is homeomorphic to $${\mathbb {R}}^{\text {rank}\,G-1}$$, where $$A$$ is a maximal real split torus and $$M$$ is a maximal compact subgroup that normalizes $$N$$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures. 

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4. BiLipschitz homogeneous hyperbolic nets

   https://doi.org/10.54330/afm.152404  

   Bishop, Christopher J (June 2024, Annales Fennici Mathematici)

   We answer a question of Itai Benjamini by showing there is a \(K< \infty\) so that for any \(\epsilon >0\), there exist \(\epsilon\)-dense discrete sets in the hyperbolic disk that are homogeneous with respect to \(K\)-biLipschitz maps of the disk to itself. However, this is not true for \(K\) close to \(1\); in that case, every \(K\)-biLipschitz homogeneous discrete set must omit a disk of hyperbolic radius \(\epsilon(K)>0\). For \(K=1\), this is a consequence of the Margulis lemma for discrete groups of hyperbolic isometries. 

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5. Largest acylindrical actions and Stability in hierarchically hyperbolic groups

   Abbott, Carolyn; Behrstock, Jason; Durham, Matthew (July 2020, Transactions of the American Mathematical Society)

   We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3–manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasi- convexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. 

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