## On random walks on wreath products.(English)Zbl 1021.60004

Let $$K$$ and $$B$$ be groups with finite generators. Considering $$H$$ as an $$H$$ set, we make a twisted and restricted wreath product $$G= K\text{ wr }H$$. Let $$\nu$$ (resp. $$\mu$$) be a symmetric probability measure whose support generates $$K$$ (resp. $$H$$) and assume that the support of $$\mu$$ contains the unit element of $$H$$. Considering these measures as measures on $$G$$, define the measure $$q =\nu\ast\mu\ast\nu$$ on $$G$$. In this article, the asymptotic behavior of $$q^{2n}(e)$$, where $$q^{2n}$$ is the $$2n$$-convolution of $$q$$ and $$e$$ is the unit element of $$G$$, is considered for various algebraic structures of groups $$K$$ and $$H$$. In formula (3.6) (p. 964) $$V(m)$$ is put to $$m$$, however it must be $$2^{m}$$, so Lemma 3.8 (p. 964), which is crucial in the following arguments, must be reexamined. In my try, it does not hold. Also many inconsistencies of symbols are to be found.

### MSC:

 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization

### Keywords:

wreath product; return probability
Full Text:

### Online Encyclopedia of Integer Sequences:

Cogrowth sequence of the group Z wr Z where wr denotes the wreath product.

### References:

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