I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat.
The Cayley graph can be found here.
My idea is to pick two points arbitrarily far to the right in the graph and pick two points that are k-distance apart then "move up one level" and pick two vertices there.
Then my geodesic triangle will have one side that can be k+1 large but the other 2 will be short (only length 1) hence by increasing k I can just increase how fat my triangle is.
Is this the correct approach and can anybody help me make it more precise?
Thanks for any help
Your method will not work, it is not possible for a triangle to have two sides of length 1 and one side of large length, because that is a violation of the triangle inequality.
Instead what you can do (referring to the same picture) is to take a point $x$ on the upper left edge, and to take two points $y,y'$ on the upper right edge, so that the distance between $y,y'$ is small, even as small as $1$. Then construct two geodesics $xy$, $xy'$: each of them starts from $x$, then goes downward in the picture, then turns back upward at the lowest level. The geodesic $xy$ turns back upward at a point $z$ at the lowest level, and the geodesic $xy'$ turns back upward at a point $z'$ also at the same lowest level. The trick is to see that $xy$, $xy'$ can be chosen so that the distance between $z,z'$ is large. Then to make the triangle fatter and fatter, choose $x$ further up to the left, and choose $y,y'$ further up to the right.
That's kind of vague, but if you can figure out the picture that I am trying to convey using much fewer than a thousand words, then it should quickly become clear.