There is a recent paper by Maxime Gheysens where he among other nice things systematically investigates the topologies on $\mathrm{Aut}(G)$ 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 $\mathrm{Aut}(G)$ 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 $\mathrm{Aut}(G)$ is purely a phenomenon in the world of “very big” groups.

$\DeclareMathOperator\Aut{Aut}$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.

There is a recent paper by Maxime Gheysens where he among other nice things systematically investigates the topologies on $\mathrm{Aut}(G)$ for any topological group $G$. As it turns out, 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 $\mathrm{Aut}(G)$ 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 $\mathrm{Aut}(G)$ is purely a phenomenon in the world of “very big” groups.

There is a recent paper by Maxime Gheysens where he among other nice things systematically investigates the topologies on $\mathrm{Aut}(G)$ 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 $\mathrm{Aut}(G)$ 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 $\mathrm{Aut}(G)$ is purely a phenomenon in the world of “very big” groups.