This is a post about topology and about groups. While theoretically it is possible and in scope for me to introduce topology from the bottom up, I suspect that you won’t get anything out of the story that I want to tell if it really is your first time seeing the definition of a topology, so I won’t. However, if you know topology and groups separately, this is a perfect place to meet topological groups, since I’m going to talk only about pretty elementary things.
A topological group is a group which is at the same time a topological space so that multiplication and inversion are continuous as maps and respectively.
An identity neighborhood is a an open subset of a topological group containing the identity, . Identity neighborhoods satisfy the following axioms.
To clarify notation, , , and . The second bullet point follows from continuity of the map and inversion, respectively, and the third bullet point follows from continuity of multiplication.
A slight weakening of these axioms suffice to characterize group topologies.
Proposition 1. Suppose that is a collection of subsets of a group satisfying the following axioms.
- We have for all .
- For each , there exists satisfying .
- For each , there exists satisfying .
- For each , there exists satisfying .
- For each and , there exists satisfying .
Then there is a group topology on where a subset is open if and only if for each there exists such that .
Proof. The claimed topology really is a topology: the defining property of open sets is clearly closed under arbitrary unions. If and , then by assumption there exists so that , so finite intersections of open sets are open.
Suppose where is open. To show that multiplication is continuous, we show that there are sets so that . Indeed, since is open, it contains for , and we choose so that and . We compute:
as required.
To show that inversion is continuous, given , take and . Then
Thus this topology makes a topological group.
Let be a group acting on a connected, simplicial graph by graph automorphisms without inversions in edges. Simplicial means that between two vertices of , there is at most one edge, and that no edge forms a loop. Without inversions in eges means that if an edge of is fixed setwise by a group element, it is fixed pointwise. A consequence of this latter condition is that the stabilizers of an edge incident to the vertices and is naturally a subgroup of the stabilizers of both and . Another is that the quotient map is a graph map.
Proposition 2. Suppose acts on a connected, simplicial graph by graph automorphisms without inversions in edges, and that for each vertex , the stabilizer is equipped with a group topology such that the following conditions hold.
- For each , conjugation by induces a homeomorphism .
- If is an edge of incident to vertices and , the inclusion of the stabilizer into and yields the same topology on , such that both inclusions are open embeddings.
Then the family satisfies the hypotheses of Proposition 1. The induced topology on is the finest group topology on such that each inclusion is an open embedding. In this topology the group acts continuously on .
Proof. We verify the hypotheses of Proposition 1. The first, that each contains the identity, is clear, as are the third and fourth, since each is a topological group. The final hypothesis, about conjugation, holds by definition. Thus we need only verify that contains an element of . Supposing that and for vertices , we will show this by induction on the distance between and . More precisely, given any geodesic edge path from to , we show that the subset of which additionally stabilizes this path pointwise belongs to , and is in fact an identity neighborhood in both and .
The base case where distance (i.e. ) holds because is a topological group.
when and are connected by an edge , the intersection is contained in the edge stabilizer .
Suppose now that the result holds for all pairs and at distance at most , we prove the statement for vertices at distance . Let be a geodesic edge path from to , and let be the initial vertex of .
By induction, the subset of the intersection which stabilizes the (possibly trivial) path is an identity neighborhood in both and . Call this subset .
Since inherits the same topology as an open subgroup of each of and , we see that is an open identity neighborhood in both and . The subset of stabilizing is clearly contained in , and the reverse is also true: if , then by assumption stabilizes the path and also , whence since is simplicial, it stabilizes the whole path.
Finally, to see that this topology is the finest group topology on such that each inclusion is an open embedding, observe that any group topology satisfying this latter hypothesis actually makes each into an identity neighborhood, so the identity map is continuous, as required. .
Proposition 2 has an immediate consequence for certain (2-categorical) colimits in the category of topological groups.
Corollary 3. Suppose is a connected graph of topological groups such that each edge-to-vertex group homomorphism is an open embedding. The fundamental group of the graph of groups is canonically a topological group.
One limitation of Corollary 3 is that it is not strong enough to topologize some of the most familiar tree-like constructions on groups. For example, I’ll close by leaving you to ponder the following.
Exercise 4. Suppose are groups. The only topology on their free product which makes the action on the natural Bass-Serre tree associated to the free product continuous is the discrete topology.