I am having a hard time understanding what almost convex means. The definition is the following:
A group $G = \langle S\rangle$ is almost convex if there exists a constant $k$ such that every two points in the sphere of radius $n$ at distance at most 2 in the Cayley graph $\Gamma(G,S)$ can be joined by a path of length at most $k$ that stays in the radius ball of lenght $n$.
I am also having a difficult time trying to solve this question: Let $G$ and $H$ be almost convex groups, show that $G \bigoplus H$ and $G * H$ are also almost convex groups.