# On the complexity of the cogrowth sequence

@article{Bell2020OnTC, title={On the complexity of the cogrowth sequence}, author={Jason P. Bell and Marni Mishna}, journal={Journal of Combinatorial Algebra}, year={2020} }

Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from $S$. We prove that the cogrowth sequence is not P-recursive when $G$ is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak. In addition, we compute the cogrowth for certain infinite families of…

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