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1. Dynamic asymptotic dimension and Matui's HK conjecture

   https://doi.org/10.1112/plms.12510  

   Bönicke, Christian ; Dell'Aiera, Clément ; Gabe, James ; Willett, Rufus ( January 2023 , Proceedings of the London Mathematical Society)

   We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side‐effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K‐theory of the ‐algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK‐conjecture.

   We also construct explicit maps from the groupoid homology groups to the K‐theory groups of their ‐algebras in degrees zero and one, and investigate their properties.

    

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2. Genus Integration, Abelianization, and Extended Monodromy

   https://doi.org/10.1093/imrn/rnz133  

   Contreras, Ivan ; Fernandes, Rui Loja ( July 2019 , International Mathematics Research Notices)

   Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in [9]. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of [6], but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

    

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3. On a random entanglement problem

   https://doi.org/10.1093/imamat/hxac031  

   Bonner, Gage ; Thiffeault, Jean-Luc ; Valkó, Benedek ( November 2022 , IMA Journal of Applied Mathematics)

   We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental groupoid of the region. We quantify entanglement of the path with the length of the appropriate element in this groupoid. Our main results are a law of large numbers and a central limit theorem for this quantity. The constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations.

    

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4. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

   https://doi.org/10.1093/imrn/rnab114  

   Duwenig, Anna ; Gillaspy, Elizabeth ; Norton, Rachael ( July 2021 , International Mathematics Research Notices)

   Abstract When the reduced twisted $C^*$-algebra $C^*_r({\mathcal{G}}, c)$ of a non-principal groupoid ${\mathcal{G}}$ admits a Cartan subalgebra, Renault’s work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r({\mathcal{G}}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid ${\mathcal{S}}$ of ${\mathcal{G}}$. In this paper, we study the relationship between the original groupoids ${\mathcal{S}}, {\mathcal{G}}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum ${\mathfrak{B}}$ of the Cartan subalgebra $C^*_r({\mathcal{S}}, c)$. We then show that the quotient groupoid ${\mathcal{G}}/{\mathcal{S}}$ acts on ${\mathfrak{B}}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map ${\mathcal{G}}\to{\mathcal{G}}/{\mathcal{S}}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on ${\mathcal{G}}/{\mathcal{S}} \ltimes{\mathfrak{B}}$. 

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5. ROOTS OF CHARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID

   Chekhov, L. ; Shapiro, M. ; Shibo, H. ( July 2022 , Uspehi matematičeskih nauk)

   We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the A_n-groupoid of upper-triangular matrices to express roots of the characteristic equation det(A−λA^T)=0, with A∈A_n, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra sl_n. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of A_n-groupoid to a A_{Sp_2m}-groupoid. 

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