Let $S$ be a closed oriented surface of genus $g\geq 2$. Let $\mathcal{T}$ be the corresponding Teichmuller space. Given a free homotopy class of closed curve $[\gamma]$ we can define the length function $l\_{\gamma}$ on $\mathcal{T}$.


It is well known that there exist $[\alpha]\neq [\beta]$ such that $l\_\alpha=l\_\beta$. 


**Q) Does there exist free homotopy classes $[\alpha]\neq [\beta]$ such that $l\_\alpha-l\_\beta=c\neq 0$ for some $c\in\mathbb{R}$?** 



Length functions on Teichmuller space with constant difference
Let me first prove an easier statement, namely that there are no loops $c\_1, c\_2$ on $S$ such that $tr(\rho(c\_1)) - tr(\rho(c\_2))=a$ for some nonzero constant $a$ and all discrete and faithful representations $\rho: \pi\_1(S)\to SL(2, {\mathbb R})$. Consider the representation variety 
$$
Rep(S)=Hom(\pi\_1(S), SL(2, {\mathbb C})). 
$$
This variety is known to be connected (proven by Bill Goldman in his PhD thesis and published sometime in 1980s). This variety is even smooth away from "reducible representations", i.e. representations whose images have centralizers in $SL(2, {\mathbb C}$ of positive dimension. 


Also, the subset
$$
F(S)=Hom\_{df}(\pi\_1(S), SL(2, {\mathbb R}))
$$
is Zariski dense (over complex numbers) in $Rep(S)$. This follows from the fact that it is an open nonempty subset in the set of real points in $Rep(S)$. 
Now, if the identity $tr(\rho(c\_1))- tr(\rho(c\_2))=a$ holds for all $\rho\in F(S)$, then it holds on the entire $Rep(S)$. On the other hand, $Rep(S)$ contains the trivial representation $\rho\_0$ and for all loops $c$ on $S$, $tr(\rho\_0(c))=2$. A contradiction.


The proof for the length functions is similar, except we have to use holomorphic functions and not quite on $Rep(S)$, but on its 2-fold branched cover. 


Recall that for a nontrivial loop $c$ on a hyperbolic surface $S={\mathbb H}^2/\Gamma$ represented by a matrix $C\in \Gamma$ under a discrete representation $\rho: \pi\_1(S)\to \Gamma< SL(2, {\mathbb R})$, the hyperbolic length of $c$ equals 
$$
\log \lambda^2\_C,
$$
where $\lambda\_C>1$ is the largest eigenvalue of $C$. 


Therefore, the identity $l(c\_1)-l(c\_2)=a\ne 0$ for all hyperbolic structures on the surface $S$ translates into the identity
$$
\lambda^2\_{C\_1}= e^a \lambda^2\_{C\_2}. 
$$
Now, let $Rep'(S)$ denote the Zariski subset of $Rep(S)$ consisting of representations such that both matrices $\rho(c\_1), \rho(c\_2)$ have eigenvalues of the multiplicity one, i.e. are different from $\pm 1$. 
On $Rep(S')$ we have multivalued (actually, each function takes only two values) holomorphic functions $\lambda^2\_1, \lambda^2\_2$ obtained by analytic continuation of the real-analytic functions $\lambda^2\_{C\_1}, \lambda^2\_{C\_2}$ defined on $F(S)$. To make these functions single-valued we pass to a 4-fold analytic cover $Rep''(S)\to Rep'(S)$. We obtain single-valued lifts $\mu\_k$ of $\lambda\_k, k=1,2$ to $Rep''(S)$. Now, it follows that the identity 
$$
\mu\_1= e^a \mu\_2
$$ 
still holds on $Rep''(S)$. The trouble is that the trivial representation is not in $Rep'(S)$. However, there is a real curve $\alpha(t)$, $t\in (0,1]$ in 
$Rep'(S)$ such that $\lim\_{t\to 0}\alpha(t)=\rho\_0$, the trivial representation. 


Then, lifting $\alpha$ to a curve $\beta$ in $Rep''(S)$, we again obtain a contradiction:
$$
\lim\_{t\to 0} \mu\_k \beta(t)= 1, k=1, 2, 
$$
contradicting the identity $\mu\_1= e^a \mu\_2$, $a\ne 0$. 


I can chase references if anybody cares. 


