I'm working on a problem that involves an amenable group acting on some set by bijections. Initially, I assumed the group was discrete and the set was countable, however I realized that the arguments I have generalize to the group being a locally compact second countable topological group, and that the set may be uncountably infinite. I have not been able to find an example of such a group or action however, largely due to my unfamiliarity with topological groups. In addition, the group action needs to satisfy some additional requirements in order for my results to apply. Imprecisely, there needs to be "plenty" of elements in the group which fix arbitrary finite subsets of the set it's acting on. The precise requirement implies that the group cannot be compact or abelian.
What are some examples of such groups that naturally act by bijections on some uncountable set? An example where the uncountable set is some nice subset of the reals (i.e. $(0,1)$, $[0,1]$, $[0,1)$, all of $\mathbb{R}$) would be especially nice, however any example would be nice. Thanks!
