Let $G$ be a locally compact topological group with unity $e$ and left Haar measure $m$. Let also $g\in G$ be a given element and $U$ a neighborhood (of compact closure) of $e$.

I am interested to get another, smaller neighborhood $V$, contained in $U$, with the conjugation-invariance property $gVg^{-1}=V$ (with respect to the given element $g$).

This is immediate for Abelian $G$. Also, it is easy if the order of $g$ is finite, say $o(g)=n$, because then we can just take $V:=\cap_{j=0}^{n-1} g^jUg^{-j}$.

If $o(g)=\infty$, this may fail, as relatively easy examples can show. However, I would be satisfied by a quazi-invariance property: for any $0<\varepsilon<1$, find some $V\subset U$ with $m(V\cap gVg^{-})> (1-\varepsilon)m(V)$.

When do we have the first, or the approximative second, invariance property? What generally well-known properties, conditions ensure it and what groups (and for what kind of elements, besides of course having $o(g)=\infty$) or perhaps neighborhoods, (if the property depends on that) are such that it fails?