If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this for a while but I am not sure how to prove this (I am not too happy about proving things are $k$-thin in general hence my other questions)
I know that the Cayley graph "looks like" a line with the vertices for all the integers and that say on the top we have lines going from $x$ to $x+3$ and on the bottom that we have lines going from $x$ to $x+5$.
Now I am confused as how to show that any triangle is $k$-thin? I am slightly confused as to what a geodesic triangle actually looks like here?
Idea If I take a geodesic triangle between the points say $0,3-2$ then two of these sides have length $1$ and the other has length $2$ (the side connecting $-2$ to $0$ as we have to go via $-5$ then this triangle is $1$-thin but I have no idea how to generalise this argument to all geodesic triangle
I know that this type of question is poorly suited here (as I can't draw pictures) but some help with this would be great.
Thanks for any help