Let $S$ be a surface of negative Euler characteristic (for simplicity let's assume $S$ to be closed), and let $\mathcal{M}(S)$ denote the moduli space of hyperbolic surfaces homeomorphic to $S$. 


Given $X \in \mathcal{M}(S)$, let $L(X)= l\_{1} \leq l\_{2} \leq l\_{3} \leq...$ denote the increasing sequence of lengths of all closed geodesics on $X$. This is the so-called "unmarked length spectrum". It is known that $L(X)$ does not determine $X$ up to isometry; indeed, if $\mathcal{V} \subset \mathcal{M}(S)$ denotes the set of all surfaces $X$ for which there exists $Z \neq X$ with $L(X) = L(Z)$ (in this case $X$ and $Z$ are called $\textit{iso-spectral}$), then it is a theorem of Wolpert that $\mathcal{V}$ is a real-analytic subvariety of $\mathcal{M}(S)$ with positive dimension. On the other hand, $\mathcal{V}$ also has positive codimension, so "generically" $L(X)$ does determine the metric. 


McKean proved that any subset $W \subset \mathcal{M}(S)$ consisting of pairwise iso-spectral surfaces must be finite, and Buser gave an explicit upper bound on $|W|$ which depends only on the genus of $S$. 


Let $d\_{T}( , )$ denote the projection of the Teichmuller metric to $\mathcal{M}(S)$, and similarly $d\_{WP}( , )$ for the Weil-Petersson metric. 


$\textbf{Question 1}:$ Let $D\_{T}(S):= \sup \left\{ d\_{T}(X,Y) : \hspace{2 mm} X,Y \in \mathcal{M}(S), X \hspace{2 mm} \mbox{iso-spectral to} \hspace{2 mm} Y \right\}$, and define $D\_{WP}(S)$ similarly. Is $D\_{T} < \infty$? How about $D\_{WP}$? (of course if $D\_{T}$ is finite then so is $D\_{WP}$). 


$\textbf{Question 2:}$ Let $LS(X)$ denote the subsequence of $L(X)$ consisting only of lengths corresponding to $\textbf{simple}$ closed geodesics. Is it the case that $LS(X)= LS(Y)$ implies $L(X)= L(Y)$? 


$\textbf{Regarding Question 1:}$ I suspect that the answer is "no" for both $D\_{T}$ and $D\_{WP}$, but I don't have an argument in mind, nor am I very familiar with what is known about iso-spectrality in general. I'll remark that there exists fairly general/algebraic methods for constructing iso-spectral surfaces with arbitrarily large genus, such as the following due to Sunada: 


Let $\phi: \pi\_{1}(X) \rightarrow G$ be a surjective homomorphism on to a finite group $G$, and let $H\_{1}, H\_{2}$ be $\textit{almost conjugate}$ subgroups of $G$, meaning that 


$$ | [g] \cap H\_{1}| = | [g] \cap H\_{2} |, \forall g \in G, $$


where $[g]$ denotes the conjugacy class of $g$. Then the covering spaces of $X$ associated to the subgroups $\phi^{-1}(H\_{1})$ and $\phi^{-1}(H\_{2})$ will be iso-spectral. Perhaps, one can construct iso-spectral surfaces that are very far apart in $\mathcal{M}(S)$ by carefully choosing $G$, $\phi$ and $X$. 


$\textbf{Regarding question $2$:}$ It is known that the converse is false. That is, $L(X)= L(Y)$ does not imply that $LS(X)= LS(Y)$ (see the following preprint of Maungchang: <http://arxiv.org/abs/1111.4317>). The proof uses Sunada's construction described above.


Thank you very much for reading. 



Teichmuller distance between isospectral riemann surfaces
As a partial answer to question one: The diameter of $M(S)$ with respect to the Weil--Petersson metric is finite. So $D\_{WP}$ is finite. 


