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1. Cataclysms for Anosov representations

   https://doi.org/10.1007/s10711-022-00721-7  

   Pfeil, Mareike ( August 2022 , Geometriae Dedicata)

   In this paper, we construct cataclysm deformations for$$\theta $$$\theta$-Anosov representations into a semisimple non-compact connected real Lie groupGwith finite center, where$$\theta \subset \Delta $$$\theta \subset \Delta$is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston’s cataclysms on Teichmüller space and Dreyer’s cataclysms for Borel-Anosov representations into$$\mathrm {PSL}(n, \mathbb {R})$$$\mathrm{PSL}(n,R)$. We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for$$\theta $$$\theta$-Anosov representations.

    

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2. Right-angled Artin groups as normal subgroups of mapping class groups

   https://doi.org/10.1112/S0010437X21007417  

   Clay, Matt ; Mangahas, Johanna ; Margalit, Dan ( August 2021 , Compositio Mathematica)

   We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara. 

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3. Carnot metrics, dynamics and local rigidity

   https://doi.org/10.1017/etds.2021.116  

   CONNELL, CHRIS ; NGUYEN, THANG ; SPATZIER, RALF ( February 2022 , Ergodic Theory and Dynamical Systems)

   Abstract This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces. 

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4. Log-canonical coordinates for symplectic groupoid and cluster algebras

   https://doi.org/10.1093/imrn/rnac101  

   Chekhov, L. ; Shapiro, M. ( January 2022 , International mathematics research notices)

   Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the A_n groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A_3 and A_4 in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for A_n via sequences of cluster mutations in the special A_n-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. 

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5. Largest hyperbolic actions and quasi-parabolic actions in groups

   https://doi.org/10.1142/S1793525322500066  

   Abbott, Carolyn R. ; Rasmussen, Alexander J. ( January 2022 , Journal of Topology and Analysis)

   The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the “best” hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit the largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have the largest hyperbolic actions, including for instance many 3-manifold groups and most mapping class groups. Our proofs use the quasi-trees of metric spaces of Bestvina–Bromberg–Fujiwara, among other tools. In addition, we give a complete characterization of the poset of hyperbolic actions of Anosov mapping torus groups, and we show that mapping class groups of closed surfaces of genus at least two have hyperbolic actions which are comparable only to the trivial action. 

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