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1. Sub-Riemannian Limit of the Differential Form Heat Kernels of Contact Manifolds

   https://doi.org/10.1093/imrn/rnaa270  

   Albin, Pierre; Quan, Hadrian (October 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the $$\eta $$-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative $$\eta $$-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological. 

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2. The Borel complexity of the space of left‐orderings, low‐dimensional topology, and dynamics

   https://doi.org/10.1112/jlms.70024  

   Calderoni, Filippo; Clay, Adam (November 2024, Journal of the London Mathematical Society)

   Abstract We develop new tools to analyze the complexity of the conjugacy equivalence relation , whenever is a left‐orderable group. Our methods are used to demonstrate nonsmoothness of for certain groups of dynamical origin, such as certain amalgams constructed from Thompson's group . We also initiate a systematic analysis of , where is a 3‐manifold. We prove that if is not prime, then is a universal countable Borel equivalence relation, and show that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . We also prove that if is the complement of a nontrivial knot in then is not smooth, and show how determining smoothness of for all knot manifolds is related to the L‐space conjecture. 

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3. Asymptotic expansion of smooth interval maps.

   https://doi.org/10.24033/ast.1110  

   Rivera-Letelier, Juan (January 2020, Asterisque)

   Sylvain Crovisier; Raphael Krikorian; Carlos Matheus; Samuel Senti. (Ed.)

   We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be cal- culated in various different ways. A consequence is that several natural notions of nonuniform hyperbolicity coincide. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unimodal maps. This also solves a conjecture of Luzzatto in dimension 1. Combined with a result of Nowicki and Przytycki, these considerations imply that several natural nonuniform hyperbolicity conditions are invariant under topo- logical conjugacy. Another consequence is for the thermodynamic formalism: A non- degenerate smooth map has a high-temperature phase transition if and only if it is not Lyapunov hyperbolic. 

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4. Quantum representations and monodromies of fibered links.

   Renaud Detcherry, Efstratia Kalfagianni (July 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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5. Cosets of monodromies and quantum representations

   Renaud Detcherry, Efstratia Kalfagianni (April 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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