Narrow Results

Given an Artin system (A,S), a conjecture of Tits states that the subgroup A⁽²⁾ of A generated by the squares of the generators in S is subject only to the obvious commutator relations between generators. In particular, A⁽²⁾ is a right-angled Artin group. We prove this conjecture for a class of infinite type Artin groups, called locally reducible Artin groups, for which the associated Deligne complex has a CAT(0) geometry. We also prove that for any special subgroup A_T of A, A⁽²⁾∩A_T=(A_T)⁽²⁾.

The author previously introduced groups BV and as braided versions of groups introduced by Richard J. Thompson, and as having connections to braided tensor categories. Here we show that BV and are finitely presented by exhibiting finite presentations for them. We show that these groups are "Artinifications" of the older Thompson groups V and by showing that presentations for V and can be obtained from presentations for BV and by adding relations that trivialize the squares of the generators of the latter groups.

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight "forbidden" graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right-angled Artin group to contain no hyperbolic surface subgroups.

We prove that for one of these "forbidden" subgraphs P₂(6), the right-angled Artin group A(P₂(6)) is a subgroup of a (right-angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).

We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is, the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterize the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show how Aut(G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decomposition of Aut(G) into a subgroup of locally conjugating automorphisms by the subgroup stabilizing a certain lattice of "admissible subsets" of the vertices of C. We then characterize those graphs for which Aut(G) is a product (not necessarily semi-direct) of two such subgroups.

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

We examine the construction of Huang and Osajda that was used in their proof of the biautomaticity of Artin groups of almost large type. We describe a slightly simpler variant of that biautomatic structure, with explicit descriptions of a few small examples, and we examine some of the properties of the structure. We explain how the construction can be programmed within the GAP system.

The mod 4 braid group, ${{𝒵}}_n$, is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements, $P{B}_n^2$. Kordek and Margalit proved ${{𝒵}}_n$ is an extension of the symmetric group by ${\mathbb{Z}}_2^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}$. For $n\ge 1$, we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying ${{𝒵}}_n$. We show this cocycle is the mod 2 reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show that it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for $P{B}_n^2$.

A general result of Epstein and Thurston implies that all link groups are automatic, but the proof provides no explicit automaton. Here we show that the groups of all torus links are groups of fractions of so-called Garside monoids, i.e., roughly speaking, monoids with a good theory of divisibility, which allows us to reprove that those groups are automatic, but, in addition, gives a completely explicit description of the involved automata, thus partially answering a question of D. F. Holt.

Computation of fundamental groups of Galois covers recently led to the construction and analysis of Coxeter covers of the symmetric groups [L. H. Rowen, M. Teicher and U. Vishne, Coxeter covers of the symmetric groups, J. Group Theory8 (2005) 139–169]. In this paper we consider analog covers of Artin's braid groups, and completely describe the induced geometric extensions of the braid group.

We show that the twisted conjugacy problem for extra-large Artin groups and for the class of CLTTF Artin groups introduced by J. Crisp in Ref. 18 is solvable, relative to length preserving automorphisms. We also show that these groups have infinite number of such twisted conjugacy classes and twisted conjugacy of elements of a standard parabolic subgroup which is invariant under the automorphism in the group, implies twisted conjugacy already in the parabolic subgroup.