In a previous post, I introduced a set of axioms for identity neighborhoods in topological groups. In general, many of the familiar categorical constructions for topological groups appear to me to be extremely poorly understood so far. Some of this appears to me to be intrinsic to the subject, but plenty of it does not.

For example, in this post I want to describe a collection of group topologies on the free product $G * H$ of topological groups $G$ and $H$, and determine which one makes the free product into the categorical coproduct in the category of topological groups.

Free Products, A Refresher

If $S$ is a set (called an “alphabet”), the “free monoid” $S^*$ over $S$, also sometimes called the collection of “words” or “strings” in $S$ is the collection of finite sequences $s_1s_2\ldots s_n$, where each $s_i \in S$. Concatenation of strings is an associative, typically noncommutative binary operation on $S^*$ with a two-sided identity, namely the empty string. (The operation, its associativity and identity are what make $S^*$ deserve the appellation “monoid”.)

In general, $S^*$ is not a group, because there is no inverse to concatenation. However, when $G$ and $H$ are groups, there is an equivalence relation on $(G \cup H)^*$ whose quotient is a group. If $\omega_1$ and $\omega_2$ are strings with equivalence classes $[\omega_1]$ and $[\omega_2]$ respectively, the group operation is

$[\omega_1][\omega_2] = [\omega_1\omega_2].$

In other words, concatenation of strings descends to become the group operation.

Here is the equivalence relation: if $g_1 g_2 = g_3$ in $G$ or $h_1 h_2 = h_3$ in $H$, we set

$\omega_1 g_1 g_2 \omega_2 \sim \omega_1 g_3 \omega_2 \quad\text{and}\quad \omega_1 h_1 h_2 \omega_2 \sim \omega_1 h_3 \omega_2,$

where $\omega_1$ and $\omega_2$ are arbitrary strings. Also if $1_G$ and $1_H$ are the identity elements of $G$ and $H$ respectively, we also set

$\omega_1 1_G \omega_2 \sim \omega_1\omega_2 \quad\text{and}\quad \omega_1 1_H \omega_2 \sim \omega_1\omega_2.$

Two strings are equivalent if one can be transformed into the other by a finite collection of moves.

The quotient of $(G \cup H)^*$ by the equivalence relation $\sim$ generated by these four operations is a group $G * H$, which we call the free product of $G$ and $H$, where the inverse of a word $s_1\ldots s_n$ is represented by $s_n^{-1}\ldots s_1^{-1}$, where the inversion of each letter happens in the group $G$ or $H$ as the case may be.

This equivalence relation is extremely nice from a theory of computation standpoint, by the way. Every word can be represented uniquely by a word which cannot be shortened by any of these four operations, and such words can be recognized by a finite state automaton with four states: START, G, H and REJECT. Only the REJECT state is not an accept state, and the start state is START. From START, every nonidentity element of $G$ takes you to G, and every nonidentity element of $H$ takes you to H. At G nonidentity elements of $H$ take you to H and conversely at H, nonidentity elements of $G$ take you to G. All other possibilities take you to REJECT.

Provided you know how to compute in $G$ and $H$, you can adapt the above automaton to help you do computations in $G * H$. A “greedy algorithm”, for example, which works by seeing where in the word we move to REJECT and performing corresponding one of our four basic operations which reduces word length and trying again will run in more or less linear time.

Free products of nontrivial groups $G$ and $H$ are always infinite, nonabelian (i.e. the group operation is noncommutative) and contain canonical copies of both $G$ and $H$ as (non-normal) subgroups.

The free product satisfies a “universal property”: If $K$ is a group and $G \to K$ and $H \to K$ are homomorphisms, there exists a unique homomorphism $G * H \to K$ such that the restriction of this homomorphism to the canonical copy of $G$ or $H$ respectively gives back the homomorphisms we started with.

The Bass-Serre Tree

The abstract group $G * H$ acts on a tree called the Bass-Serre tree with one orbit of edges and two orbits of vertices. The edges of the tree correspond to elements of the group, and vertices of the tree correspond to cosets of the canonical copies of $G$ and $H$. The edge $g$ connects the vertices $g G$ and $g H$. The action of $G * H$ is multiplication in the labels. The stabilizer of the vertex $g H$, for example, is the conjugate $g H g^{-1}$ of the subgroup $H$ in $G * H$.

It is perhaps not immediate to see that the graph so described is a tree; this is part of a very beautiful subject called Bass-Serre theory and named for Jean Pierre Serre, who wrote a book Trees on the subject, and Hyman Bass who was his grad student at the time, contributed to the theory from the beginning and continued to develop the subject for some time after.

At the time of this writing, Serre is 99 years old and posted a math paper to the arXiv within the past two years.

Some Topologies

In the previous post alluded to earlier, we found a set of axioms characterizing group topologies in terms of their identity neighborhoods. In short, identity neighborhoods are closed under finite intersections, inversion, multiplication, and conjugation.

Supposing $G$ and $H$ are topological groups, we are interested in group topologies on $G * H$ such that the canonical inclusions of $G$ and $H$ are embeddings. To this end, we must allow any identity neighborhood in $G$ or $H$ to be at least contained in an identity neighborhood of $G * H$. The challenges, however, are finite intersections between identity neighborhoods in $G$ with those in $H$, as well as ensuring that conjugates of identity neighborhoods are identity neighborhoods.

Thinking about it this morning, the problem feels very similar to the question of topologizing Cartesian products of spaces. I have three families of sets, $\mathcal{B}$, $\mathcal{C}$ and $\mathcal{P}$, each of which satisfies the axioms and induces a different group topology on $G * H$.

First, a little notation: let $T$ be the Bass-Serre tree from above and $v$ a vertex, and let $\mathop{\mathrm{Stab}}(v)$ be its stabilizer. As we saw above, $v$ corresponds to a coset of the form $g G$ or $g H$, and we may choose a unique preferred representative of this coset by requiring that the reduced word representing $g$ either ends with an letter in $H$ in the case of $gG$ or with a letter in $G$ in the case of $gH$. Call this element $g_v$. In all cases below, let $U_v$ denote an identity neighborhood in $\mathop{\mathrm{Stab}}(v)$.

$\mathcal{P} = \left\{ \bigcup_{v \in T} U_v : U_v = \mathop{\mathrm{Stab}}(v) \text{ for all but finitely many } v \right\}$

I called this $\mathcal{P}$ because its definition is reminiscent to me of the product topology on a Cartesian product.

$\mathcal{C} = \left\{ \bigcup_{v \in T} U_v : U_v = g_v U_G g_v^{-1} \text { or } g_v U_H g_v^{-1} \text{ for fixed } U_H \subset H \text{ and } U_G \subset G \right\}$

The identity neighborhoods feel a bit like “cylinders”, since you fix once and for all $U_H$ and $U_G$ and just conjugate them around, hence why I chose the name $\mathcal{C}$.

$\mathcal{B} = \left\{ \bigcup_{v \in T} U_v \right\}$

Beyond being open, there are no constraints here; one picks a separate identity neighborhood for each $v$. This topology felt most similar to the box topology on a Cartesian product, hence the name $\mathcal{B}$.

A little elementary set theory should convince you that each element of $\mathcal{P}$ contains an element of $\mathcal{C}$, and that each element of $\mathcal{C}$ is an element of $\mathcal{B}$, so of the three resulting topologies, $\mathcal{B}$ is the finest, meaning it has the most open sets, while $\mathcal{P}$ is the coarsest, having the fewest open sets.

Since the intersection of $\mathop{\mathrm{Stab}}(v)$ with $\mathop{\mathrm{Stab}}(w)$ is trivial (i.e. just the identity) when $v \ne w$, you can work term by term to see that if $U$ and $V$ belong to one of the collections above, then so do $U^{-1}$, $g U g^{-1}$ for any $g$ and $U \cap V$. Likewise, for each $U$ there is a $W$ such that $W^2 \subset U$. Therefore each of the collections $\mathcal{P}$, $\mathcal{C}$ and $\mathcal{B}$ produces a group topology on $G * H$.

For each of these topologies, the canonical inclusions of $G$ and $H$ are embeddings. Also for each of these topologies, one can show that if $\Phi\colon G \to K$ and $\Psi\colon H \to K$ are continuous homomorphisms of topological groups, the induced homomorphism $G * H \to K$ is continuous.

The idea I have for this is to begin by supposing that $g \in G$ and $h \in H$, and considering an identity neighborhood $\Phi(g)\Psi(h)W$ in $K$. By continuity of multiplication in $K$, there are identity neighborhoods $V_g$ and $V_h$ in $K$ such that $\Phi(g)V_g \Psi(h)V_h \subset \Phi(g)\Psi(h)W$. By continuity of $\Phi$ and $\Psi$, there are identity neighborhoods $U_g \subset G$ and $U_h \subset H$ such that $\Phi(g U_g)\Psi(g U_h) \subset \Phi(g) V_g \Psi(h)V_h$. We can then use $U_g$ and $U_h$ to define elements of the desired family of identity neighborhoods. Some careful induction should complete the proof.

Differentiating the topologies

Exercise. The canonical homomorphism $G * H \to G \times H$, where the latter has the product topology, is continuous for the $\mathcal{C}$-topology (and hence the $\mathcal{B}$-topology) but in general not for the $\mathcal{P}$-topology.

The point, you should convince yourself in doing the exercise, is that the preimage of the basic identity neighborhood $U \times V$ contains the element of $\mathcal{C}$ determined by $U$ and $V$, but contains no element of $\mathcal{P}$.

It’s fairly obvious that the $\mathcal{B}$-topology is in general genuinely finer than the $\mathcal{C}$-topology, but I’m not aware of a universal construction that lets you readily see this. That is, I don’t know of a topological group $K$ not isomorphic to $G * H$ equipped with homomorphisms from the factors for which the abstract homomorphism $G * H \to K$ is continuous for the $\mathcal{B}$-topology but not for the $\mathcal{C}$-topology. If you can think of one, email me!, I’d love to hear about it.

Proposition. The $\mathcal{B}$-topology is the finest group topology on $G * H$ for which the inclusion maps $G \to G * H$ and $H \to G * H$ are continuous. It is therefore the coproduct of $G$ and $H$ in the category of topological groups.

Proof. Suppose that $\tau$ is a group topology on $G * H$ as in the statement. We show that an arbitrary identity neighborhood in $\tau$ contains a $\mathcal{B}$-identity neighborhood. This will show that $\tau$-open sets are $\mathcal{B}$-open, completing the proof.

If $U$ is a $\tau$-identity neighborhood, we know that $U_G = U \cap G$ and $U_H = U \cap H$ are identity neighborhoods, as is $gUg^{-1}$ for any $g$. More generally, if $g = g_v$ for $v \in T$, notice that $g_v^{-1} U g_v \cap U \ni 1$ is an identity neighborhood, so we conclude that $U \supset g_v U_v g_v^{-1}$ for some identity neighborhood $U_v \subset \mathop{\mathrm{Stab}}(v)$. This constructs an element of $\mathcal{B}$ contained in $U$. $\QED$