If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?

Question

Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a fixed conjugacy class of $x$. If there exist a simple closed loop $z$ (non-trivial) with $i(x,z)\neq 0$ , where $i(\ , \ )$ denotes the geometric intersection number. Then is it true that $i(x*y,z)\neq 0$?

One can prove that if $y=x$ (upto fixed base point homotopy) , then $i(x,z)\neq 0$ implies $i(x^2,z)\neq 0$ , because the hyperbolic axis correspond to the translation $[x],[x^2]\in \pi_1(S_g)$ in $\mathbb H^2$ are same. Also the reason I've chosen $z$ to be a simple closed loop, becasue I think $x*y$ cannot be a simple closed loop here.

I'm unable to find some proof/counter-example. Although my intuition says that it should be true but I would be happy if some one give me a counter-example with a good (general) explanation.

Answer 1

If $x$ is simple, then this is true. Here is an outline of the proof.

1) As $x$ and $y$ are freely homotopic, $y=x^g$ for some $g$. As $x$ is simple, $A_x$ and $A_y$ are disjoint.

2) Consider the Theorem 7.38.3 of Beardon's "Geometry of discrete group." Observe that $g$ is the common perpendicular, therefore $\epsilon =+1$.

3) This implies if length of $x$ increases, so does length of $x*y$ (in Teichnuller space).

4) Take right twists along the curve $z$. This will increase the length of $x$ (as $i(x,z)\neq 0$) and hence must increase the length of $x*y$ which implies $i(x*y,z)\neq=0$.

As long as $A_x$ and $A_y$ are disjoint, this proof will work. The other case is when $A_x\cap A_y\neq \emptyset.$ Then $x$ is not simple and their product formula is given by Theorem 7.38.3 of Beardon's "Geometry of discrete group," where $v_2$ will be a lift of the self-intersection point. Now there are two cases:

1) If $cos\theta$ is positive at $v_2$ then the above arguments hold true.

2) If $cos\theta$ is negative, it is the same question as thismentioned by Ian Agol for the following reason. Take the self intersection point and consider the two branches of the curves starting and ending at this intersection point. Name them $x$ and $y$. If $y$ is a power of $x$ then you are done by your observation. If not then you get the case $n=1$ of the above question.

Answer 2

This is a counter exampleb of my question. (proposed by Dan Margalit)