Abstract
A group $G$ is simple if, for any elements $f,g\in G$ with $g\neq 1$, $f$ is a product of finitely many conjugates of $g$. If this finite number is bounded by $N$ (independently on $f$ and $g$) we call $G$ $N$-boundedly simple. We will show that many groups (known to be simple) acting on a line or a circle, such as various Thompson, Higmann-Thompson, or Neretin groups, are $N$-boundedly simple with explicit bound for $N$. Presented results are a joint work with Światosław R. Gal from Wrocław/Jerusalem/Haifa.