Mauricio Gutierrez

BRAIDS AND THE NIELSEN-THURSTON CLASSIFICATION

The well-known connection between braids and mapping classes on the n-punctured disc can be exploited to decide, using braid-theoretic techniques, the place of a given isotopy class in the Nielsen-Thurston classification as reducible or irreducible, and in the latter case, as periodic or pseudo-Anosov.

On the automorphisms of a graph product of abelian groups

We study the automorphisms of a graph product of finitely generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition Aut* W = (Inn W x Out(0) W) x Aut(1) W. We also give a number of applications, some of which are geometric in nature.

A GENERATING SET FOR THE AUTOMORPHISM GROUP OF A GRAPH PRODUCT OF ABELIAN GROUPS

We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurences paper [11].

Free simplicial groups and the second relative homotopy group of an adjunction space

(If (YX) is a pair of spaces, then 7rnz(YX) is a ;rrtX-crossed module.) Suppose that X is a connected CW-complex, and that Y is obtained from X by attaching 2-cells. Whitehead [9] showed that in this case, x2(xX) is a free x,X-crossed module, i.e., if {c,} are the elements of ;71#‘,X) corresponding to the 2-cells of Y-X, and if (g,} are elements of another n,X-crossed module G with ag,=ac,, then there is a unique homomorphism of crossed modules h : 7t2(Y,X) + G for which h(c,) =g,. Whitehead’s proof of this theorem is rather difficult and geometric (see [l] for a more modern exposition of Whitehead’s proof). Brown and Higgins [2] have shown that this theorem follows from their 2-dimensional generalization of the van Kampen theorem, in the context of ‘double groupoids’. Ratcliffe [7], using his study of free and projective crossed modules, was able to give a more algebraic proof. The purpose of the present paper is to give a completely algebraic proof of this theorem. We will obtain the theorem through a study offree simplicial groups, which are models for loop spaces (and are therefore useful for computing (absolute) homotopy groups).

RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

We show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the To property. Our results build on results by Droms, Laurence and Radcliffe.

Automorphisms of Right-Angled Coxeter Groups

If (𝑊,𝑆) is a right-angled Coxeter system, then Aut(𝑊) is a semidirect product of the group Aut∘(𝑊) of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut∘(𝑊) is a semidirect product of Inn(𝑊) by the quotient Out∘(𝑊)=Aut∘(𝑊)/Inn(𝑊). We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence 1→Inn(𝑊)→Aut(𝑊)→Out(𝑊)→1 and consequently, all group extensions 1→𝑊→𝐺→𝑄→1 are trivial.

Comultiplications on free groups and wedges of circles

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group F . We consider individual comultiplications on F and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of F . For a comultiplication m of F we define a subset ∆m ⊆ F of quasi-diagonal elements which is basic to our investigation of associativity. The subset ∆m can be determined algorithmically and contains the set of diagonal elements Dm. We show that Dm is a basis for the largest subgroup Am of F on which m is associative and that Am is a free factor of F . We also give necessary and sufficient conditions for a comultiplication m on F to be a coloop in terms of the Fox derivatives of m with respect to a basis of F . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group AutF on the set of comultiplications of F . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.

COMULTIPLICATION ON MONOIDS

A comultiplication on a monoid S is a homomorphism m:S→S∗S (the free product of S with itself) whose composition with each projection is the identity homomorphism. We investigate how the existence of a comultiplication on S restricts the structure of S. We show that a monoid which satisfies the inverse property and has a comultiplication is cancellative and equidivisible. Our main result is that a monoid S which satisfies the inverse property admits a comultiplication if and only if S is the free product of a free monoid and a free group. We call these monoids semi-free and we study different comultiplications on them.