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1. Dichotomy and Measures on Limit Sets of Anosov Groups

   https://doi.org/10.1093/imrn/rnad188  

   Lee, Minju; Oh, Hee (October 2023, International Mathematics Research Notices)

   Abstract Let $$G$$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $$\Gamma <G$$, we show that a $$\Gamma $$-conformal measure is supported on the limit set of $$\Gamma $$ if and only if its dimension is $$\Gamma $$-critical. This implies the uniqueness of a $$\Gamma $$-conformal measure for each critical dimension, answering the question posed in our earlier paper with Edwards [13]. We obtain this by proving a higher rank analogue of the Hopf–Tsuji–Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups. 

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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. Sarnak’s spectral gap question

   Kelmer, Dubi; Kontorovich, Alex; Lutsko, Christopher (December 2023, Journal d'Analyse Mathématique)

   We answer in the affirmative a question of Sarnak’s from 2007, confirming that the Patterson–Sullivan base eigenfunction is the unique square-integrable eigenfunction of the hyperbolic Laplacian invariant under the group of symmetries of the Apollonian packing. Thus the latter has a maximal spectral gap. We prove further restrictions on the spectrum of the Laplacian on a wide class of manifolds coming from Kleinian sphere packings. 

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4. Pseudo-Anosov subgroups of general fibered 3–manifold groups

   https://doi.org/10.1090/btran/157  

   Leininger, Christopher; Russell, Jacob (August 2023, Transactions of the American Mathematical Society, Series B)

   We show that finitely generated and purely pseudo-Anosov subgroups of fundamental groups of fibered 3–manifolds with reducible monodromy are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. Combined with results of Dowdall–Kent–Leininger and Kent–Leininger–Schleimer, this establishes the result for the image of all such fibered 3–manifold groups in the mapping class group. 

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5. Manhattan curves for hyperbolic surfaces with cusps

   https://doi.org/10.1017/etds.2018.124  

   KAO, LIEN-YUNG (July 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not. 1993 (7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z. 228 (4) (1998), 745–750] for convex cocompact Fuchsian representations. 

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