Finite Groups Of Type Research Articles

Helly meets Garside and Artin

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

A note on the groups of finite type and the Hartman–Mycielski construction

Ando, Matsuzawa, Thom, and Törnquist have resolved a problem by Sorin Popa by constructing an example of a Polish group of unitary operators with the strong operator topology, whose left and right uniform structures coincide, but which does not embed into

Finite graph product closure for a conjecture on the BNS-invariant of Artin groups

We work with a conjecture on the BNS-invariant of Artin groups stated by Almeida and Kochloukova. We show that the class of Artin groups that satisfy this conjecture is closed under finite graph products. As a consequence, we show that the conjecture is true for all Artin groups of finite type and other subclasses.

On the Poincaré-Hopf Theorem for Functionals Defined on Banach Spaces

Abstract Let X be a reflexive Banach space and f : X → ℝ a Gâteaux differentiable function with f′ demicontinuous and locally of class (S)+. We prove that each isolated critical point of f has critical groups of finite type and that the Poincaré-Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result.

RANDOMIZED METHODS IN ARTIN GROUPS OF FINITE TYPE

We investigate the use of random algorithms for the solution of the conjugacy problems for certain types of Artin groups of finite type including the braid groups.

Solution of the generalized conjugacy problem for words in pure subgroups of Artin groups of finite type

The notion of pure subgroup of an Artin group of finite type is introduced. The decidability of the generalized conjugacy problem for pure subgroups of Artin groups of finite type is proved.

Convexes divisibles III

Let Δ 0 be a group of finite type and F Δ 0 ⊂ Hom ( Δ 0 , PGL ( R m ) ) be the subset of faithful representations for which there exists a properly convex Δ 0 -invariant open subset Ω in P ( R m ) such that the quotient Δ 0 \\ Ω is compact. Koszul has proved in [J.L. Koszul, Déformation des connexions localement plates, Ann. Inst. Fourier 18 (1968) 103–114] that this subset F Δ 0 is open. We describe the closure of F Δ 0 . As a consequence, we show that this subset F Δ 0 is closed if and only if the virtual center of Δ 0 is trivial. This condition is satisfied if and only if F Δ 0 contains a strongly 1 1 I.e. the restriction of the representation to any subgroup of finite index of Δ 0 is still irreducible. irreducible representation.

EXTRACTION OF ROOTS IN GARSIDE GROUPS

ABSTRACT V. B. Styshnev showed in [9] that the existence of n -th roots for a braid is decidable. Garside groups have been introduced in [2] and [3] as a natural proper generalization of Artin groups of finite type. We have to construct a new proof to extend Styshnev's decidability result to Garside groups, as several specific properties of braids used in [9] fail in our case. We show that, under the assumption of a finiteness property of conjugacy, the problem is decidable.

ON THE FIRST CARTAN INVARIANT FOR THE FINITE GROUP OF TYPE

ABSTRACT The first Cartan invariant for the finite group of type over a field of elements with or is calculated in this note.

MINIMAL INDUCTIVE SYSTEMS OF MODULAR REPRESENTATIONS FOR NATURALLY EMBEDDED ALGEBRAIC AND FINITE GROUPS OF TYPE A

The article is devoted to the classification of the minimal and minimal nontrivial inductive systems of modular representations for naturally embedded algebraic and finite groups of type A and related locally finite groups. It occurs that the minimal systems consist of the trivial representations for the relevant groups and the minimal nontrivial ones are connected with Frobenius twists of the standard representations and their duals. These results are applied to the description of the maximal ideals in group algebras of the locally finite groups SL ∞ and SU ∞ in describing characteristic. It is also proved that for an arbitrary classical algebraic group, the restriction of an irreducible module with highest weight large enough to a naturally embedded finite Chevalley group of the same type, but a smaller rank contains the regular module.

MINIMAL INDUCTIVE SYSTEMS OF MODULAR REPRESENTATIONS FOR NATURALLY EMBEDDED ALGEBRAIC AND FINITE GROUPS OF TYPE A

The article is devoted to the classification of the minimal and minimal nontrivial inductive systems of modular representations for naturally embedded algebraic and finite groups of type A and related locally finite groups. It occurs that the minimal systems consist of the trivial representations for the relevant groups and the minimal nontrivial ones are connected with Frobenius twists of the standard representations and their duals. These results are applied to the description of the maximal ideals in group algebras of the locally finite groups SL ∞ and SU ∞ in describing characteristic. It is also proved that for an arbitrary classical algebraic group, the restriction of an irreducible module with highest weight large enough to a naturally embedded finite Chevalley group of the same type, but a smaller rank contains the regular module.

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg +↦g −, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG.

Admissible subgroups of full ergodic groups

AbstractLet G be a countable group of automorphisms of a Lebesgue space (X, m) and let [G] be the full group of G. For a pair of countable ergodic subgroups H1 and H2 of [G], the following problem is considered: when are the full subgroups [H1] and [H2] conjugate in the normalizer N[G] = {g ∈ Aut X: g[G]g-1 = [G]} of [G]. A complete solution of the problem is given in the case when [G] is an approximately finite group of type II and [H] is admissible, in the sense that there exists an ergodic subgroup [H0] of [G] and a countable subgroup Γ ⊂ N[H0] consisting of automorphisms which are outer for [H0], such that [H0] ⊂ [G] and the full subgroup [Ho, Γ] generated by [H0] and Γ coincides with [G].

Diffeomorphisms of manifolds with finite fundamental group

We show that the group D ( M ) \mathcal {D}(M) of pseudoisotopy classes of diffeomorphisms of a manifold of dimension ≥ 5 \geq 5 and of finite fundamental group is commensurable to an arithmetic group. As a result π 0 ( D i f f M ) {\pi _0}(Diff\,M) is a group of finite type.

Unsolvability of the occurrence problem in Artin groups of finite type

Unsolvability of the occurrence problem in Artin groups of finite type

Occurrence problem in Artin groups of finite type

Occurrence problem in Artin groups of finite type