Equivalence proven between two torsion invariants for flat vector bundles.
problem Equivalence of Igusa-Klein and Bismut-Lott torsion invariants for flat vector bundles.
method Reduction to trivial flat line bundles using Artin's induction theorem.
result Igusa-Klein and Bismut-Lott torsion invariants are equivalent for flat vector bundles with finite holonomy.
We observe an inductive structure in a large class of Artin groups and exploit this information to deduce the Farrell-Jones isomorphism conjecture for several classes of Artin groups of finite real, complex and affine types.
We study monoids generated by Zariski-van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C^3 listed by Sekiguchi (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaussian and, hence, are neither G…
Reduces conjecture for Artin groups to simpler cases.
problem Proving K(π,1) for Artin groups with specific spherical parabolics. method Reduces to simpler cases, uses injective metric spaces, combinatorial convexity, and Bestvina-type inequalities.
result Deduces K(π,1) conjecture for specific Artin groups. Artin groups of hyperbolic type are boundary amenable and have rigid properties.
problem Characterizing rigidity and measure equivalence properties of Artin groups.
method Analyzing boundary amenability, measure equivalence, and fixed set graphs.
result Measure equivalent Artin groups of hyperbolic type have isomorphic fixed set graphs.
In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of …
The study shows that certain Artin groups cannot contain hyperbolic manifold groups.
problem Proving that certain Artin groups cannot contain hyperbolic manifold groups.
method Elementary proof, using the Virtual Fibering Conjecture and Belegradek's splitting theorem.
result Right-angled Artin groups cannot contain finite volume hyperbolic 3-manifold groups.
There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …
We give a short proof of the following theorem of Sang-hyun Kim: if A(Γ) is a right-angled Artin group with defining graph Γ, then A(Γ) contains a hyperbolic surface subgroup if Γ contains an induced subgraph Cˉn for some n≥5, where Cˉn denotes the complement graph of an n-cycle. Furthe…
Right-angled Artin groups have unique quasi-isometry classes when measure equivalent.
problem Characterizing when right-angled Artin groups are measure equivalent.
method Proving measure equivalence implies quasi-isometry and using geometric properties of cube complexes.
result Measure equivalence of right-angled Artin groups implies quasi-isometry and geometric properties.
We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.
Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.
problem Overcoming singularities in the Schoen-Yau proof for arbitrary dimensions.
method Inductive scheme combining shielding principle, conformal blow-up, and Cheeger-Naber bound.
result Proof of positive mass theorem in arbitrary dimensions.
We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A(G) with connected defining graph G. We use this to determine when two points in an asymptotic cone of A(G) are separated by a cut-point. As an application, we show that if G does not d…
The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.
problem Understanding the structure and properties of orbifold braid groups.
method Definitions and proofs of fibration theorems, short exact sequences, and poly-virtually free structures.
result The pure orbifold braid groups have poly-virtually free structure, generalizing classical braid groups.
The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.
problem Understanding geometric finiteness in mapping class groups and constructing new examples.
method Examined several constructions of subgroups and determined conditions for geometric finiteness.
result Provides new examples of parabolically geometrically finite and reducibly geometrically finite subgroups.
For a finite simplicial graph Γ, let A(Γ) denote the right-angled Artin group on Γ. Recently Kim and Koberda introduced the extension graph Γe for Γ, and established the Extension Graph Theorem: for finite simplicial graphs Γ1 and Γ2 if Γ1 embeds into Γ2e as an induced subgraph then A(Γ1) emb…
Study of Coxeter diagrams and Artin-Tits groups, focusing on normalisers and wall intersections.
problem Understanding normalisers of parabolic subgroups in Artin-Tits groups and their connections to Coxeter diagrams.
method Analyzing hyperplane arrangements, Coxeter groups, and wall-and-chamber structures.
result Complexified hyperplane complement is a K(π,1) space for normalisers of parabolic subgroups in finite-type Coxeter diagrams.
New right-angled Artin subgroups found in Artin groups.
problem Finding large right-angled Artin subgroups in Artin groups.
method Examining centers of irreducible spherical special subgroups and their powers.
result Conjecture verified for certain classes of Artin groups, leading to hyperbolic surface subgroup conclusions.
No free lunch theorems suggest inductive biases are needed, but we show neural networks prefer low-complexity data.
problem The need for inductive biases in machine learning.
method Analysis of Kolmogorov complexity and neural network behavior on various datasets.
result Neural networks prefer low-complexity data, suggesting inductive biases are not always necessary.
New proofs show smallest non-cyclic quotients for braid and mapping class groups.
problem Classifying smallest non-cyclic quotients of braid and mapping class groups.
method Elementary proofs without Bertrand-Chebyshev theorem.
result Smallest non-cyclic quotients for braid and mapping class groups identified.
An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.
We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges over all right-angled Artin groups. This is in contrast with the cases where A is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))| < 5. We…
The paper introduces a Deligne complex for Artin monoids and studies its properties.
problem Understanding geometric structures associated with Artin monoids.
method Constructing a Deligne complex for Artin monoids and analyzing its properties.
result The Deligne complex for Artin monoids is contractible and can be embedded into the Deligne complex for the corresponding Artin group.
In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow the L--move methods to prove …
Artin groups not free of infinity are shown to have finite centers.
problem Characterizing Artin groups with finite centers.
method Reduced clique-cube complexes and actions on them.
result Artin groups not free of infinity have finite centers, and are trivial in many cases.
The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.
problem Investigate properties of Artin monoid Cayley graphs.
method Show quasi-isometry to modified Deligne complex, address infinite diameter conjecture.
result Prove conjecture about infinite diameter for Artin groups containing specific subgroups.
Reduces conjecture to tree-based Artin groups.
problem Proving K(π,1)-conjecture for all Artin groups. method Actions on Bestvina complexes of Garside groupoids.
result New classes of Artin groups satisfying the conjecture.
Artin groups get K(π,1)-conjecture proof for tree and cyclic diagrams.
problem Proving K(π,1)-conjecture for Artin groups. method New approach to Artin groups, focusing on tree and cyclic diagrams.
result Established K(π,1)-conjecture for specific Artin groups. The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.
problem Characterizing and classifying 4-manifolds using Artin presentations and triangle groups.
method Utilizing triangle groups to find Artin presentations that present the trivial group and determining 4-manifolds with specific properties.
result Identified all Artin presentations on two generators that present the trivial group and all smooth, closed, simply-connected 4-manifolds with specific properties.
We will give a new proof for the Gromov's theorem on almost flat manifolds, which is an inductive proof on dimension.
The study proves conjecture for specific Artin groups.
problem Proving conjecture about Artin groups' properties.
method Analyzing Artin groups associated to triangle-free graphs and cones over square-free bipartite graphs.
result Proves conjecture for specific Artin groups.
New normal subgroups found in mapping class groups.
problem Finding normal subgroups in mapping class groups.
method New windmill apparatus tailored to projection complexes.
result Normal subgroups isomorphic to non-free right-angled Artin groups.
Even Artin groups generalize right-angled Artin groups by allowing the labels in the defining graph to be even. In this paper a complete characterization of quasi-projective even Artin groups is given in terms of their defining graphs. Also, it is shown that quasi-projective even Artin groups are realizable by K(pi,1) …
Many 2D Artin groups are residually finite.
problem Residual finiteness of 2D Artin groups.
method Showed these groups split as free products with amalgamation or HNN extensions of finite rank free groups.
result Many 2D Artin groups are residually finite.
Uniqueness of quasi-roots explored in right-angled Artin groups.
problem Uniqueness of quasi-roots in right-angled Artin groups.
method Introducing quasi-roots and studying their uniqueness.
result Uniqueness of quasi-roots established in right-angled Artin groups.
This paper characterizes a specific type of twisted Artin groups embedded in knot groups.
problem Embedding twisted right-angled Artin groups in knot groups.
method Defined and characterized twisted right-angled Artin groups through mixed graphs and Klein bottle relations.
result Completely determined which twisted right-angled Artin groups can be embedded in knot groups.
Introduces stability conditions and their connection to Artin groups.
problem Understanding the relationship between stability conditions and Artin groups.
method Explains the connection between Bridgeland stability conditions and the K(π,1) conjecture. result Establishes a link between stability conditions and Artin groups.
For every integer l bigger than one, we find elements x and y in the mapping class group of an appropriate orientable surface S, satisfying the Artin relation of length l. That is, xyx... = yxy..., where each side of the equality contains l terms. By direct computations, we first find elements x and y in Mod(S) satisfy…
Classifies braids with positive Artin presentations and their fundamental groups.
problem Classifying braids with positive Artin presentations and understanding their fundamental groups.
method Analyzing framed, closed pure n-braids in the 3-sphere to determine if they represent positive Artin presentations.
result Closed, pure n-braids B' in the 3-sphere that represent positive Artin presentations are strongly invertible, and some 3-manifolds do not admit such presentations.
The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
problem Understanding the relationship between graph structures and algebraic properties of right-angled Artin groups.
method Analyzing the defining and extension graphs of right-angled Artin groups.
result Discovers connections to geometric group theory and complexity theory.
We describe a simple locally CAT(0) classifying space for extra extra large type Artin groups (with all labels at least 5). Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra large type Artin groups are acylindrically hyperbolic. Together with Property RD …
Artin groups have finite stature based on vertex groups.
problem Understanding the finite stature of Artin groups.
method Criteria for finite stature based on vertex groups, applied to triangle Artin groups.
result Residual finiteness of Artin groups expanded.
Artin groups of types F4 and H4 are not commensurable with D4.
problem Determining commensurability between Artin groups of spherical type.
method Realized the abstract commensurator of D4 as the extended mapping class group of a torus with three punctures; found the automorphism group and described torsion elements. result Artin groups of types F4 and H4 are not commensurable with D4. Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.