We establish sufficient conditions implying semistability and connectivity at infinity properties for CAT(0) cubical complexes. We use this, along with the geometry of cubical K(π, 1)’s to give a complete description of the higher connectivity at infinity properties of right angled Artin groups. Among other things, this determines which right angled Artin groups are duality groups. Applications...

We will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, τ -categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers. 0.1 There exists a bijection between equivalence classes of Krull-Schmidt categories C with additive gen...

we prove that for $p>7$ there are‎ ‎[‎ ‎p^{4}+2p^{3}+20p^{2}+147p+(3p+29)gcd (p-1,3)+5gcd (p-1,4)+1246‎ ‎] ‎groups of order $p^{8}$ with exponent $p$‎. ‎if $p$ is a group of order $p^{8}$‎ ‎and exponent $p$‎, ‎and if $p$ has class $c>1$ then $p$ is a descendant of $‎p/gamma _{c}(p)$‎. ‎for each group of exponent $p$ with order less than $‎p^{8} $ we calculate the number of descendants of o...

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...

It is known that the recently discovered representations of the Artin groups of type An, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type Dn and En which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I1 and I2 with I...

We determine the epimorphisms A → W from the Artin group A of type Γ onto the Coxeter group W of type Γ, in case Γ is an irreducible Coxeter graph of spherical type, and we prove that the kernel of the standard epimorphism is a characteristic subgroup of A. This generalizes an over 50 years old result of Artin. AMS Subject Classification: Primary 20F36; Secondary 20F55.

which for every finite Galois extension L/K induces an isomorphism invL/K : H (Gal(L/K), L×) ∼ −→ 1 [L:K]Z/Z via composition with the inflation map Inf : H2(Gal(L/K), L×)→ H2(Gal(Ksep/K),Ksep×), and we showed that invK is canonical and natural in K; see Theorem 30.25 In this lecture we will use the invariant map to define the local Artin homomorphism and prove the local version of Artin recipro...

Bestvina-Brady groups arise as kernels of length homomorphisms GΓ → Z from right-angled Artin groups to the integers. Under some connectivity assumptions on the flag complex ∆Γ, we compute several algebraic invariants of such a group NΓ, directly from the underlying graph Γ. As an application, we give examples of finitely presented Bestvina-Brady groups which are not isomorphic to any Artin gro...

We construct four families of Artin-Schelter regular algebras of global dimension four. This is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1 and whose Ext-algebra satisfies certain “generic” conditions. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Keller’s...

A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an ArtinSchelter regular algebra is Artin-Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin-Schelter regular algebras of global dimension four in a seque...