# Almost Convex Groups

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.

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So, what do you know about almost convex groups? Do you know the definition? –  Gerry Myerson May 7 '12 at 4:31
I Suggest you that you read chapter seven from Geometric group theory, an introduction by Clara Loh –  Babak Miraftab May 7 '12 at 4:57
I do not understand the definition. –  Rhonda May 7 '12 at 6:41
I think the question was not whether you understand the definition (you already said in the question that you don't) but whether you know it. –  joriki May 7 '12 at 10:47
@joriki has it right - maybe if you produce the definition someone will be able to help you understand and/or use it. –  Gerry Myerson May 7 '12 at 13:00