Surface homeomorphism transitively permutes the boundary curves

Question

Let $h: S\to S$ be a surface homeomorphism. What does it mean to say that $h$ transitively permutes components of $\partial S$, and where does this terminology come from? Is this somehow related to a transitive group action?

Answer 1

It is indeed very closely related to a transitive group action. But first you need some simple facts from topology.

The first fact needed is that $h(\partial S) = \partial S$ and $h$ restricts to a self-homeomorphism of $\partial S$.

The second fact is that the $h$ image of each component of $\partial S$ is a component of $\partial S$. Furthermore, $h$ induces a bijection from the set of components of $\partial S$ to itself, so if we denote the component decomposition of $\partial S$ as $\partial S = \{C_1,...,C_K\}$, then there is a permutation $\pi$ of $\{1,...,K\}$ so that $h(C_i)=C_{\pi(i)}$ (these facts are true of any self-homeomorphism).

Third, to say that $h$ acts transitively on the components of $\partial S$ means that for each $i,j \in \{1,...,K\}$ there is an exponent $n \in \mathbb Z$ such that $h^n(C_i)=C_j$.

The connection with group actions is that (as with any self-bijection) the bijection that $h$ induces on the set $\{C_1,...,C_K\}$ induces an action of the group $\mathbb Z$ on $\{C_1,...,C_K\}$, where $n \cdot C_i = C_j$ is equivalent to $h^n(C_i)=C_j$. So $h$ transitively permutes the set of components $\{C_1,...,C_K\}$ if and only if the action of $\mathbb Z$ on that set is transitive.

Answer 2

Too long for a comment, but I don't study surfaces, so this is my interpretation based on what those words usually mean in other contexts.

My assumption would be the following. A surface homeomorphism induces a homeomorphism of the boundary. Thus if $\Gamma$ is a subgroup of surface homeomorphisms, $\Gamma$ acts on $\partial S$. This action descends to an action on path components, so $\Gamma$ acts on $\pi_0 (\partial S)$. Then I would assume $\Gamma$ acts transitively on the components of $\partial S$ means that $\Gamma$ acts transitively on $\pi_0(\partial S)$.

Now if $h$ is a single surface homeomorphism, I would assume that $h$ transitively permutes the components of $\partial S$ means that the subgroup $\Gamma =\langle h\rangle$ acts transitively on the components of $\partial S$.

Equivalently, for all components $X$ and $Y$ of $\partial S$, there is some integer power $n$ such that $h^n$ carries $X$ to $Y$.