The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:51:48Z http://mathoverflow.net/feeds/question/42940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42940/the-growthrate-of-the-homology-of-h-m-otimes-a-n-for-a-dg-bimodule-m The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$ Vera 2010-10-20T20:47:45Z 2012-03-26T02:13:29Z <p>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..</p> http://mathoverflow.net/questions/42940/the-growthrate-of-the-homology-of-h-m-otimes-a-n-for-a-dg-bimodule-m/92220#92220 Answer by Dylan Thurston for The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$ Dylan Thurston 2012-03-26T02:13:29Z 2012-03-26T02:13:29Z <p>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 <a href="http://front.math.ucdavis.edu/1012.1032" rel="nofollow">http://front.math.ucdavis.edu/1012.1032</a>. In particular, it's easy to give examples where the growth rate is linear, although I'm sure there are also more elementary constructions.</p> <p>(There are also other, earlier constructions for the braid group, by Khovanov-Seidel and Khovanov-Thomas. I don't know if they explicitly stated the fact about growth rates, but it follows directly from their results.)</p>