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Cannon introduced the notion of almost convexity for a Cayley graph of a group to develop effective algorithms for understanding the geometry of the Cayley graph. Though wide classes of groups are almost convex, there are a number of weaker notions of almost convexity satisfied by yet more groups. Mihalik and Tschantz introduced the notion of tame 1-combings for Cayley complexes, and there are connections between the very strongest notions of tame combings and the strongest notions of almost convexity. We answer questions about possible further connections between these two notions by showing that there are groups which satisfy some of the strongest tame combing conditions which do not satisfy even the weakest non-trivial almost convexity conditions. Examples include some Baumslag-Solitar groups and Thompson's group F. This is joint work with Susan Hermiller, Melanie Stein and Jennifer Taback.