Cayley's Vision and Semantic Unfolding in Combinatorial Group Theory: The Topological and Geometric Role of the Non-Commutative Free Group in Two Generators

The first abstract definition of the notion of a finite group was given by Cayley in 1854. Most significantly, in the same work he demonstrates that every abstract group is isomorphic to a group of permutations, a result that is well-known presently as "Cayley's theorem'' in group theory.

The important thing to notice is that a group may act on itself, and this can happen in two ways. First, a group may act on itself by left multiplication; Second, a group may act on itself by conjugation. Only the first of the above different actions is free and transitive, in general. It is precisely, the self-action of a group on itself by left multiplication that has been depicted by Cayley, giving rise to the homonymous theorem. Since, this self-action is realized as a group homomorphism from the group to the group of permutations of its underlying set of elements, every group is realized isomorphically as a subgroup of some symmetric group. The challenge in geometrizing this notion is to envision a group as the group of symmetries of a geometric object. In particular, is it possible to geometrize the notion of a group itself? The answer comes from considering the realization of a group not as a group acting on a set, but as a group acting on a graph, called the Cayley graph of this group.

Combinatorial Perspective on Group Theory by von Dyck

This perspective refers to the characterization of a group in terms of generators and relations, or equivalently, in terms of a presentation of a group. In this combinatorial setting, there appears first the fundamental notion of a free group, which is characterized by a certain number of generators without any additional defining relations beyond the existence of an inverse for each generator according to the general requirements of the notion of a group. Therefore, there immediately arises the fact that every morphism of a set of free generators onto a set of elements of any group, defines a homomorphism of the free group into this group. The effect of this result is that a presentation of a finitely generated group is expressed as a quotient group of a certain free group of finite rank with respect to the congruence relations among the generators.

Non-Commutative Free Group in Two Generators

The Cayley graph of the free group in two non-commuting generators is a tree whose vertices have valence four. In this sense, the free group on two non-commuting generators is characterized as the automorphism or symmetry group of a 4-valent tree. Clearly, a 4-valent tree is simply connected since there are no cycles, and can be also endowed with a metric making it into a geometric space.

The considered metric is the path metric, that is, the metric imposed on the set of its vertices such that the distance between two vertices is the length of the shortest path made through edges connecting these two vertices. Thus, we immediately deduce that a group together with a generating set may be thought of as a metric space in such a way that its actions take place via isometries.

The action of the free group on its Cayley graph, i.e. on the corresponding 4-valent tree, is an action without any fixed points, thus it is a free group action. It turns out that this property characterizes uniquely a group as a free group: If a group acts freely on a tree, then this group is a free group.

Therefore, the qualification that a group is free is equivalent to the condition that this group acts freely on a tree, and in this way, it is completely characterized by its free action as a group on a tree. The consequence of this equivalence is the Nielsen-Schreier theorem, stating that any subgroup of a free group is also free.

The Free Group as an Autogenetic Group and the Apeiron

If we recall Spinoza's adage about what qualifies something as free:

Only that thing is free which exists by the necessities of its own nature, and is determined in its actions by itself alone;

We realize that what is called the free group mathematically, bears precisely the semantics of autogenetic constellatory self-unfolding, and thus, can be equivalently called as the "autogenetic group in two non-commuting generators'', where these generators express temporal actions.

In this manner, if the free non-commutative group in two generators is realized in terms of its temporal actions, then it is characterized uniquely as the symmetry group of a 4-valent tree and conversely.

This tree deciphers the universal form of constellatory self-unfolding of temporal actions in a simply-connected geometric way. The growth of this 4-valent tree is boundless, and as such, it is not directly experiential, thus, it is associated with the Apeiron.

Any other type of unfolding is subsumed in the above form, due to the mathematical fact that a free group in any number of generators bigger than two is included as a subgroup of the latter.

Galois Covering Action of the Free Group

The 4-valent tree being simply connected and amenable to the metric structure of its natural path metric qualifies as a geometric space whose group of symmetries is the free group in two non-commuting generators. Taking into account that this group action on the 4-valent tree is free and transitive, it qualifies as a Galois action.

Thus, the free group in two non-commuting generators is manifested as a group of covering automorphisms of the 4-valent tree, in its pertinent role as a universal covering space that annihilates the fundamental group of its quotient by this action.

In this respect, and in relation to our general focus on semantic unfolding bearing a temporal connotation, we consider the free group in two non-commuting generators in its homotopical realization as the fundamental group of a bouquet of two unlinked circles (equivalently called a 2-rose, or a rose with two petals), whose universal simply-connected geometric covering space is precisely the 4-valent tree.

Commutator of Free Group and the Borromean Link

The Borromean link is encoded as the commutator element of two unlinked circles, and as such, it constitutes the basic element of the fundamental group of a bouquet of two circles, identified with the free group in two non-commuting generators. The annihilation of this fundamental group takes place by means of the universal simply connected geometric covering space of the bouquet, identified as a 4-valent tree, whose Galois group of covering automorphisms is precisely this free group.

Abelianization

According to Hurewicz, who modelled the notion of Abelianization in the reduction of the fundamental group to the first homology group of a topological space:

The concept of homotopy is a mathematical formulation of the intuitive idea of a continuous transition between two geometrical configurations. The concept of homology gives a mathematical precision to the intuitive idea of a curve bounding an area or a surface bounding a volume ... the basic process of homology theory consisting in decomposing a space into smaller pieces with simpler homology structure has no counterpart in homotopy theory.