Related papers: The $K(\pi,1)$-conjecture implies the center conje…

Multifraction reduction is a new approach to the word problem for Artin-Tits groups and, more generally, for the enveloping group of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite…

Suppose a residually finite group $G$ acts cocompactly on a contractible complex with strict fundamental domain $Q$, where the stabilizers are either trivial or have normal $\mathbb{Z}$-subgroups. Let $\partial Q$ be the subcomplex of $Q$…

In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(\pi,1) conjecture for…

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that the conjecture is true when a finite non-abelian $p$-group $G$ has…

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. The…

We observe that, for each positive integer n > 2, each of the Artin groups of finite type A_n, B_n=C_n, and affine type \tilde A_{n-1} and \tilde C_{n-1} is a central extension of a finite index subgroup of the mapping class group of the…

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction…

We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the Shapovalov form on…

We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number…

The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality…

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both…

Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint…

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the…

In this paper, we compute Tian's $\alpha_{m,k}^{K\times K}$-invariant on a polarized $G$-group compactification, where $K$ denotes a maximal compact subgroup of a connected complex reductive group $G$. We prove that Tian's conjecture (see…

We prove that the universal covering $\Wi M$ of an $n$-dimensional closed spin PSC manifold $M$ whose fundamental group $\pi_1(M)$ is a right-angled Artin group has macroscopic dimension $\dim_{mc}\Wi M\le n-2$. This confirms Gromov's…

This paper is a sequel of [A.M. Cohen, L. Paris, {\it On a theorem of Artin}, J. Group Theory {\bf 6} (2003), 421--441]. Let $A$ be an Artin group, let $W$ be its associated Coxeter group, and let $CA$ be its associated coloured Artin…

We extend the notions of quasi-monomial groups and almost monomial groups, in the framework of supercharacter theories, and we study their connection with Artin's conjecture regarding the holomorphy of Artin $L$-functions.

Several authors have remarked the convenience of understanding the different notions of center appearing in Geometry (centroid of a set of points, incenter of a triangle, center of a conic and many others) as functions. The most general way…

We show that Neretin groups have no non-trivial invariant random subgroups. These groups provide first examples of non-discrete, compactly generated, locally compact groups with this property.

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH).…