Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close to the identity by conjugation). Has this set $M$ been studied/used anywhere? I don't have a precise question here, this is more of a general reference request.

I am particularly interested in the case that $G$ is totally disconnected and locally compact (t.d.l.c.) and in understanding $M \cap U$ where $U$ is an open compact subgroup, but I'd be interested to see what is known for connected groups as well. One can show that $M$ is non-trivial if $G$ is t.d.l.c., compactly generated and has no non-trivial compact normal subgroups, but I don't have any sense of the structure of $M$ in this case, either topologically or algebraically. There has been some work done on contraction subgroups, that is, sets of elements that are sent towards the identity by repeatedly applying a fixed endomorphism of $G$, but as far as I know, $M$ can be non-trivial even if all the (inner) automorphisms have trivial contraction groups.

Perhaps there is some representation-theoretic interpretation? Certainly if $G$ is linear, then $M$ consists of unipotent elements, though I can't see what the conditions would be for $M$ to contain all unipotent elements. What role might $M$ have in infinite-dimensional unitary representations?