# The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$

Suppose you have an DG-algebra $A$, and a DG-bimodule $M$ over $A$. Under which conditions will the rank of the bimodules $H_*(M^{\otimes_A n})$ will grow exponentially in terms of $n$? Here $M^{\otimes_A n}$ is the A-bimodule $M\otimes_A M\otimes_A\cdots\otimes_A M$ (with $n$ terms). Anything, even a counterexample would be interesting..

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I don't know in general, but in Heegaard Floer theory, there are bimodules naturally associated to a mapping class of a surface self-homeomorphism. The rank of $H_*(M^{\otimes_A n})$ grows exponentially iff the underlying mapping class group element is pseudo-Anosov. See our paper at http://front.math.ucdavis.edu/1012.1032. In particular, it's easy to give examples where the growth rate is linear, although I'm sure there are also more elementary constructions.