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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

Being almost convex means that balls in the Catkey graph don't have 'horns', i.e. long subsets that almost touch each other at the tips (spare close in the Cayley graph) but are far apart traveling in the ball. If someone folded the Earth (like Inception) so that Shanghai was only a mile in the air above London, the cities would be close (a mile apart), but if one is restricted to walking on the surface of the earth, the distance is very long.

For the second part, just change each coordinate one at a time. It won't make you leave the ball.

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