Recently I have been reading the book A Primer on Mapping Class Groups by Benson Farb and Dan Margalit. Let $S_g=$ Closed surface of genus $g$, and $Mod(S_g)$ denoted the corresponding mapping class group.
Let $f \in Mod(S_g)$, define mapping torus $M_f= \dfrac{S_g \times [0,1]}{(x,0) \sim(\phi(x),1)}$, where $[\phi]=f$, more precisely $\phi$ represenatative for $f$ and $\phi \in Homeo^+(S_g)$.
If $f$ is pseudo-Anosov, then $M_f$ is a closed hyperbolic three-manifold.
My question is the following I want to look at simple closed geodesic in $M_f$. Are they infinitely many in $M_f$?
My attempt is the following.
I am using the fact For any pseudo-Anosov homeomorphism $\phi$ of a compact surface $S$, the periodic points of $\phi$ are dense in $S$. Let $P_{\phi}$ be the set of all periodic points for $\phi$. Take $x \in P_{\phi}$ with period $n$ say then $x$ is fixed point of the pseudo-anosov map $\phi^n$, now consider the arc $x$ in $S_g \times \{0\}$ to $S_g \times \{1\}$, this is an embedded arc in $S_g \times [0,1]$ therefore in the mapping torus $M_{\phi^n}$ it will from a simple closed loop, here is also I have a question is this simple closed loop isotopic to a simple closed geodesic in $M_{\phi^n}$?....This is how I have thought to produce simple closed geodesics in $M_f$.
Thanks in advance
Theorem 9.7.2 in
Maclachlan, Colin; Reid, Alan W., The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics. 219. New York, NY: Springer. xiii, 463 p. (2003). ZBL1025.57001.
Theorem. There are infinitely many commensurability classes of (arithmetic) compact hyperbolic 3-manifolds where all closed geodesics are simple.
To connect to your question: It is now known (I. Agol, et al) that all finite volume hyperbolic 3-manifolds are virtually fibered over the circle. Therefore, WLOG, one can assume that manifolds in the above theorem are fibered over the circle.