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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?

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There's a function $\Delta$ defined by $\Delta(x)$ equals the amount by which right multiplication by $x$ scales the left Haar measure. If $G$ is compact or abelian then $\Delta$ is identically 1, but for groups like the upper triangular matrices in $GL(2,k)$ for $k$ your favourite locally compact field (e.g. the reals or the $p$-adics) this function is non-trivial. If $\Delta(g)$ is very big then the measure of $gVg^{-1}$ is much smaller than the measure of $V$, and in this case it seems to me that your quasi-invariance property cannot possibly hold. – Kevin Buzzard Aug 1 '12 at 22:52
Regarding your first question: groups where you can find such $V$ for every given $U$ and $g$ are the so-called SIN groups (where SIN stands for "small invariant enighbourhoods"). There is quite a lot known about which groups can and can't be SIN, and even a structure theorem for them due to Grosser and Moskowitz. Would further details be informative, or is this too strong a condition for what you are after? – Yemon Choi Aug 2 '12 at 0:29
Thanks, indeed. I would be very much interested in all you can tell about these SIN groups, references, description, related works etc. I am generalizing a result from already know but rather particular cases to the extent I possibly can: already the extension to the case of locally compact Abelian groups is interesting (to me, at least). But I wish to go as far as one possibly can. Therefore, a better description of this condition - which is shown to be sufficient to the generalization I am after - is certainly useful. – Szilárd RÉVÉSZ Aug 2 '12 at 7:23

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