TITLE: Is there a natural topology on the automorphism group of a topological group?
QUESTION [6 upvotes]: $\DeclareMathOperator\TAut{TAut}\DeclareMathOperator\Homeo{Homeo}$Let $G$ be a topological group, and let $\TAut(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the group of maps $f \colon G \to G$ that are simultaneously group automorphisms and self-homeomorphisms).
We wish to give $\TAut(G)$ a reasonable topology in the following sense:

$\TAut(G)$ becomes a topological group with respect to this topology.
This topology should interact with/depend on the topology of $G$ in some way, i.e. we can require that the natural action $\TAut(G) \times G \to G$ is continuous.

In the case that $G$ is compact, it is known that giving $\TAut(G)$ the compact–open topology satisfies the above conditions, where the compact–open topology has as a subbasis sets of the form $$V(C, U) = \{f \colon G \to G \mid f(C) \subseteq U\},$$ where $C, U \subseteq G$ are compact and open, respectively.
If $G$ is locally compact, we can instead give $\TAut(G)$ the $g$-topology, which has as a subbasis sets of the form $$V(K, W) = \{f \colon G \to G \mid f(K) \subseteq W\},$$ where either $K$ or $G \setminus W$ is compact.
The cases where $G$ is compact or locally compact are discussed in Dijkstra - On Homeomorphism Groups and the
Compact–Open Topology and Arens - Topologies for Homeomorphism Groups. In fact, these two papers discuss the group of self-homeomorphisms $\Homeo(X)$ for a space $X$ which is not necessarily a topological group, so the case for $G$ and $\TAut(G)$ follows from that (since $\TAut(G)$ is a subgroup of $\Homeo(G)$).
My question is, for a general topological group $G$, is there a good way to describe a topology on $\TAut(G)$ satisfying the two conditions above? We can simply say: "give $\TAut(G)$ the coarsest topology such that it becomes a topological group and the action $\TAut(G) \times G \to G$ is continuous," but I am hoping for something more explicit than that.

REPLY [8 votes]: $\DeclareMathOperator\Aut{Aut}$There is a recent paper Uniformly locally bounded spaces and the group of automorphisms of a topological group by Maxime Gheysens where he among other nice things systematically investigates the topologies on $\Aut(G)$ for any topological group $G$. On every topological group there are several natural uniform structures making translations become uniformly continuous: first of all, the left and right uniform structure, their supremum  (the upper uniform structure) and their infimum (the lower or Roelcke uniform structure). Now, as it turns out, on $\Aut(G)$ one can usefully consider:

the topology of uniform biconvergence on bounded sets with respect to the left, right, or upper uniform structure (they all give the same topology) or
the topology of uniform biconvergence on bounded sets with respect to the lower uniform structure.

In general, these two topologies are different, but they coincide for the so-called SIN groups (and even broader, for coarsely SIN groups), i.e. groups with admitting a basis of conjugation-invariant identity neighborhoods as well as for all locally compact groups. So the existence of two really different useful topologies on $\Aut(G)$ is purely a phenomenon in the world of “very big” groups.