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Christian Bönicke

Publications and source records attributed to Christian Bönicke.

14 recordsLinked to original sources
arXiv2026

Growth Conditions for Étale Groupoids: Amenability and Comparison

We investigate étale Hausdorff groupoids whose source fibres have controlled growth with respect to a length function. Our first result shows that uniform subexponential growth implies topological amenability. If the unit space is compact and metrizable and such a length satisfies a suitable uniform doubling condition, then the groupoid has weak $m$-comparison for some $m$. For minimal ample groupoids, this implies comparison and almost unperforation of the type semigroup. Under the additional assumption that the unit space has no isolated points, we deduce Matui's AH-conjecture.

math.DS↗
arXiv2024

Nuclear dimension of subhomogeneous twisted groupoid C*-algebras and dynamic asymptotic dimension

We characterise subhomogeneity for twisted étale groupoid C*-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid C*-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_\infty$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes_r D_\infty$ has finite nuclear dimension and is classifiable by its Elliott invariant.

math.OA↗
arXiv2023

Categorical approach to the Baum-Connes conjecture for étale groupoids

We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the "weakly contractible" objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of "compactly induced" algebras with respect to certain proper subgroupoids related to isotropy. The resulting "strong" Baum-Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized "Going-Down" principle, injectivity results for groupoids that are amenable at infinity, the Baum-Connes conjecture for group bundles, and a result about the invariance of $K$-groups of twisted groupoid $C^*$-algebras under homotopy of twists.

math.KT↗
arXiv2023

On the dynamic asymptotic dimension of étale groupoids

We investigate the dynamic asymptotic dimension for étale groupoids introduced by Guentner, Willett and Yu. In particular, we establish several permanence properties, including estimates for products and unions of groupoids. We also establish invariance of the dynamic asymptotic dimension under Morita equivalence. In the second part of the article, we consider a canonical coarse structure on an étale groupoid and compare the asymptotic dimension of the resulting coarse space with the dynamic asymptotic dimension of the underlying groupoid.

math.DS↗
arXiv2022

Dynamic asymptotic dimension and Matui's HK conjecture

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 $C^*$-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 $C^*$-algebras in degrees zero and one, and investigate their properties.

math.OA↗
arXiv2021

The type semigroup, comparison and almost finiteness for ample groupoids

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

math.DS↗
arXiv2020

A Going-Down principle for ample groupoids and the Baum-Connes conjecture

We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoids which are strongly amenable at infinity. This result in turn is then used to relate the Baum-Connes conjecture for an ample groupoid group bundle which is strongly amenable at infinity to the Baum-Connes conjecture for the fibres.

math.OA↗
arXiv2020

Strict comparison for $C^*$-algebras arising from Almost finite groupoids

In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $\mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup ${\rm Cu}(C_r^*(G))$ is almost divisible and ${\rm Cu}(C_r^*(G))$ and ${\rm Cu}(C_r^*(G)\otimes \mathcal{Z})$ are canonically order-isomorphic, where $\mathcal{Z}$ denotes the Jiang-Su algebra.

math.OA↗
arXiv2019

A note on crossed products of rotation algebras

We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_θ$, for any real angle $θ$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_θ$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.

math.OA↗
arXiv2019

$\mathrm{K}$-theory and homotopies of twists on ample groupoids

This paper investigates the $\mathrm{K}$-theory of twisted groupoid $\mathrm{C}^*$-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to an isomorphism between the $\mathrm{K}$-theory groups of the respective twisted groupoid $\mathrm{C}^*$-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault-Deaconu groupoids and $P$-graph algebras.

math.OA↗
arXiv2018

Going-Down functors and the Künneth formula for crossed products by étale groupoids

We study the connection between the Baum-Connes conjecture for an ample groupoid $G$ with coefficient $A$ and the Künneth formula for the K-theory of tensor products by the crossed product $A\rtimes_r G$. To do so we develop the machinery of Going-Down functors for ample groupoids. As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the Künneth formula. We also provide a stability result for the Künneth formula using controlled K-theory, and apply it to give an example of a space that does not admit a coarse embedding into a Hilbert space, but whose uniform Roe algebra satisfies the Künneth formula. As a by-product of our methods, we also prove a permanence property for the Baum-Connes conjecture with respect to equivariant inductive limits of the coefficient algebra.

math.OA↗
arXiv2018

Ideal structure and pure infiniteness of ample groupoid $C^*$-algebras

In this paper, we study the ideal structure of reduced $C^*$-algebras $C^*_r(G)$ associated to étale groupoids $G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in $C_r^*(G)$ and the open invariant subsets of the unit space $G^{(0)}$ of $G$. As a consequence, we show that if $G$ is an inner exact, essentially principal, ample groupoid, then $C_r^*(G)$ is (strongly) purely infinite if and only if every non-zero projection in $C_0(G^{(0)})$ is properly infinite in $C_r^*(G)$. We also establish a sufficient condition on the ample groupoid $G$ that ensures pure infiniteness of $C_r^*(G)$ in terms of paradoxicality of compact open subsets of the unit space $G^{(0)}$. Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: Let $G$ be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then $C_r^*(G)$ is a simple $C^*$-algebra which is either stably finite or strongly purely infinite.

math.OA↗
arXiv2017

Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of $SL_2(\mathbb{Z})$

Let $θ, θ'$ be irrational numbers and $A, B$ be matrices in $SL_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_θ\rtimes_A \mathbb{Z}$ and show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are $*$-isomorphic if and only if $θ= \pmθ' \pmod{\mathbb{Z}}$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are Morita equivalent if and only if $θ$ and $θ'$ are in the same $GL_2(\mathbb{Z})$ orbit and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Finally, we determine the Morita equivalence class of $\mathcal{A}_θ \rtimes F$ for any finite subgroup $F$ of $SL_2(\mathbb{Z})$.

math.OA↗