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Jingyin Huang

Publications and source records attributed to Jingyin Huang.

At least 19 recordsLinked to original sources
arXiv2026

Non-asphericity of strata of genus-one differentials and stability spaces

We show that when the number of zeros or poles is at least four, every connected component of the strata of differentials in genus one with prescribed zero and pole orders is not an orbifold $K(π,1)$. For quadratic differentials, this provides infinitely many counterexamples to a conjecture attributed to Kontsevich, as well as to a folklore conjecture concerning the contractibility of spaces of Bridgeland stability conditions.

math.GT↗
arXiv2026

Exotic aspherical 4-manifolds

We construct closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic. These provide counterexamples to a smooth analog of the Borel conjecture in dimension four. Our technique is to apply the `reflection group trick' of the first author to pairs of exotic 4-manifolds with boundary constructed by the second author and Piccirillo.

math.GT↗
arXiv2026

On free components of Artin and Coxeter groups

The number of connected components can be remembered by the von Neumann algebra among Artin groups, the only possible exception being the case that corresponds to the free group factor problem. In the case of Coxeter groups, this result is obtained in the absence of relatively hyperbolicity. We also discuss a specific case of the analogous problem in measure equivalence where each factor group is a product of nonabelian free groups.

math.GR↗
arXiv2026

Bestvina metric and tree reduction for $K(π,1)$-conjecture

We reduce the $K(π,1)$-conjecture for all Artin groups to properties of Artin groups whose Coxeter diagrams are trees, from which we deduce new classes of Artin groups satisfying the $K(π,1)$-conjecture. This relies on constructing actions of Artin groups on Bestvina complexes of suitable Garside groupoids.

math.GR↗
arXiv2026

From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics

We reduce the $K(π,1)$-conjecture for all Artin groups with tree Coxeter diagrams to properties of Artin groups with tripod-shaped Coxeter diagrams. Combining this reduction theorem and properties of braid groups in previous works of Charney, Crisp-McCammond, Haettel and the second named author, we deduce that the $K(π,1)$-conjecture holds for every Artin group whose spherical parabolic subgroups avoid type $D_n$ ($n \ge 4$) and the exceptional types. The reduction theorem relies on producing a ``tower'' of injective metric spaces from a single Artin group. The construction of such a tower relies on two ingredients of independent interests: a notion of combinatorial convexity and a Bestvina-type inequality, in certain injective orthoscheme complexes. These ingredients further rely on the use of structural properties of bi-Helly graphs (also known as absolute bipartite retracts) developed in joint work of the first named author with Munro.

math.GR↗
arXiv2025

A new class of affine $K(π,1)$ arrangements

We show that a certain class of affine hyperplane arrangements are $K(π,1)$ by endowing their Falk complexes with an injective metric. This gives new examples of infinite $K(π,1)$ arrangements in dimension $n>2$.

math.GR↗
arXiv2025

Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry

Let $G$ be a right-angled Artin group with $|\mathrm{Out}(G)|<+\infty$. We prove that if a countable group $H$ with bounded torsion is measure equivalent to $G$, with an $L^1$-integrable measure equivalence cocycle towards $G$, then $H$ is finitely generated and quasi-isometric to $G$. In particular, through work of Kleiner and the second-named author, $H$ acts properly and cocompactly on a $\mathrm{CAT}(0)$ cube complex which is quasi-isometric to $G$ and equivariantly projects to the right-angled building of $G$. As a consequence of work of the second-named author, we derive a superrigidity theorem in integrable measure equivalence for an infinite class of right-angled Artin groups, including those whose defining graph is an $n$-gon with $n\ge 5$. In contrast, we also prove that if a right-angled Artin group $G$ with $|\mathrm{Out}(G)|<+\infty$ splits non-trivially as a product, then there does not exist any locally compact group which contains all groups $H$ that are $L^1$-measure equivalent to $G$ as lattices, even up to replacing $H$ by a finite-index subgroup and taking the quotient by a finite normal subgroup.

math.GR↗
arXiv2025

Measure equivalence rigidity among the Higman groups

We prove that all (generalized) Higman groups on at least $5$ generators are superrigid for measure equivalence. More precisely, let $k\ge 5$, and let $H$ be a group with generators $a_1,\dots,a_k$, and Baumslag-Solitar relations given by $a_ia_{i+1}^{m_i}a_i^{-1}=a_i^{n_i}$, with $i$ varying in $\mathbb{Z}/k\mathbb{Z}$ and nonzero integers $|m_i|\neq |n_i|$ for each $i$. We prove that every countable group which is measure equivalent to $H$, is in fact virtually isomorphic to $H$. A key ingredient in the proof is a general statement providing measured group theoretic invariants for groups acting acylindrically on $\mathrm{CAT}(-1)$ polyhedral complexes with control on vertex and edge stabilizers. Among consequences of our work, we obtain rigidity theorems for generalized Higman groups with respect to lattice embeddings and automorphisms of their Cayley graphs. We also derive an orbit equivalence and $W^*$-superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.

math.GR↗
arXiv2025

Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups

We introduce a method of finding large non-positively curved subcomplexes in certain spherical Deligne complexes, which is effective for studying fillings of certain 6-cycles in spherical Deligne complexes. As applications, we show the $K(π,1)$-conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example; and the conjecture holds for all quasi-Lannér hyperbolic type Artin groups up to dimension 4. In higher dimension, we show the $K(π,1)$-conjecture for Artin groups whose Coxeter diagrams are complete bipartite (edge labels can be arbitrary), answering a question of J. McCammond.

math.GR↗
arXiv2024

Quasi-isometry classification of right-angled Artin groups II: several infinite out cases

We are motivated by the question that for which class of right-angled Artin groups (RAAG's), the quasi-isometry classification coincides with commensurability classification. This is previously known for RAAG's with finite outer automorphism groups. In this paper, we identify two classes of RAAG's, where their outer automorphism groups are allowed to contain adjacent transvections and partial conjugations, hence infinite. If $G$ belongs to one of these classes, then any other RAAG $G'$ is quasi-isometric to $G$ if and only if $G'$ is commensurable to $G$. We also show that in this case, there exists an algorithm to determine whether two RAAG's are quasi-isometric by looking at their defining graphs. Compared to the finite out case, as well as the previous quasi-isometry rigidity results for symmetric spaces, thick Euclidean buildings and mapping class groups, the main issue we need to deal with here is the reconstruction map may not have nice properties as before, or may not even exist. We introduce a deformation argument, as well as techniques from cubulation to deal with this issue.

math.GT↗
arXiv2024

Labeled four cycles and the $K(π,1)$-conjecture for Artin groups

We show that for a large class of Artin groups with Dynkin diagrams being a tree, the $K(π,1)$-conjecture holds. We also establish the $K(π,1)$-conjecture for another class of Artin groups whose Dynkin diagrams contain a cycle, which applies to some hyperbolic type Artin groups. This is based on a new approach to the $K(π,1)$-conjecture for Artin groups.

math.GR↗
arXiv2024

New Garside structures and applications to Artin groups

Garside groups are combinatorial generalizations of braid groups which enjoy many nice algebraic, geometric, and algorithmic properties. In this article we propose a method for turning the direct product of a group $G$ by $\mathbb{Z}$ into a Garside group, under simple assumptions on $G$. This method gives many new examples of Garside groups, including groups satisfying certain small cancellation condition (including surface groups) and groups with a systolic presentation. Our method also works for a large class of Artin groups, leading to many new group theoretic, geometric and topological consequences for them. In particular, we prove new cases of $K(π,1)$-conjecture for some hyperbolic type Artin groups.

math.GR↗
arXiv2024

Rigidity and classification results for large-type Artin groups

We compute the automorphism group of the intersection graph of many large-type Artin groups. This graph is an analogue of the curve graph of mapping class groups but in the context of Artin groups. As an application, we deduce a number of rigidity and classification results for these groups, including computation of outer automorphism groups, commensurability classification, quasi-isometric rigidity, measure equivalence rigidity, orbit equivalence rigidity, rigidity of lattice embedding, and rigidity of cross-product von Neumann algebra.

math.GR↗
arXiv2024

On spherical Deligne complexes of type $D_n$

Let $Δ$ be the Artin complex of the Artin group of type $D_n$. This complex is also called the spherical Deligne complex of type $D_n$. We show certain types of 6-cycles in the 1-skeleton of $Δ$ either have a center, which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center, which is a vertex adjacent to three of the alternating vertices of the 6-cycle. This will be a key ingredient in proving $K(π,1)$-conjecture for several classes of Artin groups in a companion article. As a consequence, we also deduce that certain 2-dimensional relative Artin complex inside the $D_n$-type Artin complex, endowed with the induced Moussong metric, is CAT$(1)$.

math.GR↗
arXiv2023

Indiscrete Common Commensurators

We develop a framework for common commensurators of discrete subgroups of lattices in isometry groups of CAT(0) spaces. We show that the Greenberg-Shalom hypothesis about discreteness of common commensurators of Zariski dense subgroups and lattices fails in this generality, even if one imposes strong finiteness conditions. We analyze some examples due to Burger and Mozes in this context and show that they have discrete common commensurator.

math.GR↗
arXiv2023

Lattices, Garside structures and weakly modular graphs

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural $\widetilde A_n$ shaped simplicies, including Euclidean buildings of type $\widetilde A_n$ and Cayley graphs of Garside groups and their quotients by the Garside elements. All these examples fit into the more general setting of lattices with order-increasing $\mathbb Z$-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature. We also show that several other complexes fit into this setting of lattices/lattice quotients, hence our result applies, including Artin complexes of Artin-Tits groups of type $\widetilde A_n$, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Along the way, we also clarify the relationship between categorical Garside structure, lattices with $\mathbb Z$ action and different classes of complexes studied this article. We use this point of view to describe the first examples of Garside groups with exotic properties, like non-linearity or rigidity results.

math.GR↗