# Polynomial growth of the Betti number of balls of the Cayley graphs

Consider a finitely generated group. Assume that the first Betti number of the ball of radius n in the Cayley graph is at most polynomial in n. This property is satisfied by free groups and groups of polynomial growth. Are there any others?

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The property seems to depend on the generating set. Consider the free group $\langle a,b\rangle$ with the generating set $\\{a,b,ab\\}$. The Cayley graph is a tree-graded space where pieces are triangles. A ball of radius $k$ has exponentially many triangles, each contributing 1 to the Betti number. So it seems that the growth is exponential if you use this generating set. Perhaps you should correct the question. – Mark Sapir May 4 '13 at 1:11
@Anton: My comment is equivalent to your answer. – Mark Sapir May 4 '13 at 1:18
@Mark: well, almost equivalent, almost simultaneous :) – Anton Petrunin May 4 '13 at 1:48