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}
}
  • J. Bell, M. Mishna
  • Published 21 May 2018
  • Mathematics
  • Journal of Combinatorial Algebra
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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