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1. Right-angled Artin subgroups of Artin groups

   https://doi.org/10.1112/jlms.12586  

   Jankiewicz, Kasia; Schreve, Kevin (April 2022, Journal of the London Mathematical Society)

   The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all 2-dimensional Artin groups, and for spherical Artin groups of any type other than 𝐸₆, 𝐸₇, 𝐸₈. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid. 

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2. A relative version of the Turaev–Viro invariants and the volume of hyperbolic polyhedral 3‐manifolds

   https://doi.org/10.1112/topo.12300  

   Yang, Tian (May 2023, Journal of Topology)

   Abstract We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3‐manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3‐manifolds with totally geodesic boundary. 

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3. Motivic Action on Coherent Cohomology of Hilbert Modular Varieties

   https://doi.org/10.1093/imrn/rnac126  

   Horawa, Aleksander (May 2022, International Mathematics Research Notices)

   Abstract We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $$p$$ and over $${\mathbb {C}}$$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark’s conjecture on special values of Artin $$L$$-functions and by numerical evidence in base change cases. 

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4. Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras

   https://doi.org/10.1093/imrn/rnz170  

   Xie, Zhizhang; Yu, Guoliang (August 2019, International Mathematics Research Notices)

   Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $$K_0$$-group of its group $$C^\ast$$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group. 

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5. Expected Resurgence of Ideals Defining Gorenstein Rings

   https://doi.org/10.1307/mmj/20206004  

   Grifo, Eloísa; Huneke, Craig; Mukundan, Vivek (August 2023, Michigan Mathematical Journal)

   Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy the stable Harbourne conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided that its symbolic powers are given by saturations with the maximal ideal. Although this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is Noetherian. 

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