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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?2012-05-07
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    I Suggest you that you read chapter seven from Geometric group theory, an introduction by Clara Loh2012-05-07
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    I do not understand the definition.2012-05-07
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    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.2012-05-07
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    @joriki has it right - maybe if you produce the definition someone will be able to help you understand and/or use it.2012-05-07
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    I have the definition. A group is almost convex if it has a cayley graph that is almost convex. A cayley graph is almost convex if whenever two vertices on the sphere of radius n can be joined by a path 2012-05-08
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    I thank you, and I apologize, as now that I see the definition, I, too, am having a hard time understanding what it means. Maybe the way to understand the definition is to look at a few examples of groups that are almost convex, and a few examples of groups that aren't, and get a feeling for the definition that way.2012-05-08
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    @ Gerry. I did a few examples and it made me understand the definition a little more. My problem is figuring out how to prove the direct sum of two almost convex groups is almost convex. Its driving me crazy. I have no idea how to start.2012-05-09

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