I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \subset PSL_2(\Bbb R)$. I don't understand its definition, in fact I understand it in two ways:
$\gamma$ is primitive if and only if it generates its own centralizer (i.e. commutant subgroup) $C(\gamma) \subset \Gamma$ (formally, $C(\gamma) = \langle \gamma \rangle$);
$\gamma$ is primitive if and only if the equality $\gamma = \mu ^n$ with $n >0$ implies $\gamma = \mu$ (i.e. $\gamma$ has "no radicals" in $\Gamma$).
My questions are:
is any of the two understandings above the correct one?
if both are correct, how to prove that they are equivalent?
Thank you.
There are different definitions of "primitive elements" for Fuchsian groups $\Gamma$, e.g., see the definition here. What is common to all definitions ( I hope), is that an element $\gamma_0\in \Gamma$ is primitive, iff $\gamma_0=\gamma^n$ for some $\gamma \in \Gamma$ implies $n=\pm 1$.