##
**Random walks on finite rank solvable groups.**
*(English)*
Zbl 1057.20026

Let \(\Gamma\) be a finitely generated group. Let \(p\colon\Gamma\times\Gamma\to[0,1]\) be a symmetric Markov kernel which is left-invariant, irreducible and whose support lies within bounded distance (with respect to a word metric on \(\Gamma\)) from the diagonal. We also denote by \(p\) the operator on the Hilbert space \(l^2(\Gamma)\) defined on \(f\in l^2(\Gamma)\) by the formula \(pf(x)=\sum_yp(x,y)f(y)\) and we denote by \(p^n\) the composition of \(p\) with itself \(n\) times. Let \(\delta_x\in l^2(\Gamma)\) be the characteristic function of the point \(x\). The scalar product \(p_t(x,y)=\langle p^t\delta_y,\delta_x\rangle\) can be interpreted as the probability for the random walk on \(\Gamma\) defined by \(p\) to go from \(x\) to \(y\) in time \(t\). In particular
\[
\| p^t\delta_e\|_2^2=\langle p^t\delta_e,p^t\delta_e\rangle=\langle p^{2t}\delta_e,\delta_e\rangle=p_{2t}(e,e)
\]
corresponds to the probability of return to the origin \(e\) after \(2t\) steps.

If \(f,g\) are two non-negative functions defined on positive numbers, then it will be written \(f\precsim g\) if there exist constants \(a,b>0\), such that for \(x\) large enough, \(f(x)\leq ag(bx)\). If the symmetric relation also holds, then it will be written \(f\sim g\). When a function is defined only on the integers, it is extended to the positive real axis by linear interpolation. It will be used the same name as for the original function. When using this convention for the function \(t\mapsto p_{2t}(e,e)\), the value of \(p_t(e,e)\) at an odd integer \(t\) must be interpreted as \(\tfrac12(p_{t-1}(e,e)+p_{t+1}(e,e))\).

The first example of a finitely generated group with a heat decay equivalent to \(\exp(-t^{1/3})\) was given by N. Th. Varopoulos. He showed that the heat decay on the wreath product \((\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}\) is equivalent to the expectation \(E[2^{-R_t}]\) where \(R_t\) is the random variable which counts the number of visited sites during time \(t\) for a random walk on \(\mathbb{Z}\) [see Appendix II of Bull. Sci. Math., II. Sér. 107, 337-344 (1983; Zbl 0532.60009) and the correction ibid. 108, 263-273 (1984; Zbl 0546.60008)]. The heat decay of the standard wreath product \(K\wr Q\) of two finitely generated groups \(K\) and \(Q\) with \(Q\) infinite (otherwise, the Cartesian product of \(|Q|\) copies of \(K\) is a subgroup of index \(|Q|\) in \(K\wr Q\)) behaves like \(\exp(-t^{1/3})\) if and only if \(K\) is a finite non-trivial group and \(Q\) is a finite extension of \(\mathbb{Z}\). This follows from technics developed by the authors [in Ann. Probab. 30, No. 2, 948-977 (2002; Zbl 1021.60004)].

G. Alexopoulos established the lower bound \(p_{2t}(e,e)\succsim\exp(-t^{1/3})\) for polycyclic groups [Can. J. Math. 44, No. 5, 897-910 (1992; Zbl 0762.31003)]. The main result of the paper under review is the generalization of this lower bound to the class of finitely generated solvable groups of finite Prüfer rank. Recall that a finitely generated group has Prüfer rank if there is an integer \(r\), such that any of its finitely generated subgroups admits a generating set of cardinality less or equal to \(r\).

Here are the main results of the paper under review.

Theorem 1.1. Let \(\Gamma\) be a finitely generated virtually solvable group of finite Prüfer rank. Then the heat decay in \(\Gamma\) satisfies \(p_{2t}(e,e)\succsim\exp(-t^{1/3})\).

Corollary 1.2. Let \(\Gamma\) be a finitely generated virtually solvable group of finite Prüfer rank. The heat decay of \(\Gamma\) satisfies \(p_{2t}(e,e)\sim\exp(-t^{1/3})\) if and only if \(\Gamma\) is not virtually nilpotent.

If \(f,g\) are two non-negative functions defined on positive numbers, then it will be written \(f\precsim g\) if there exist constants \(a,b>0\), such that for \(x\) large enough, \(f(x)\leq ag(bx)\). If the symmetric relation also holds, then it will be written \(f\sim g\). When a function is defined only on the integers, it is extended to the positive real axis by linear interpolation. It will be used the same name as for the original function. When using this convention for the function \(t\mapsto p_{2t}(e,e)\), the value of \(p_t(e,e)\) at an odd integer \(t\) must be interpreted as \(\tfrac12(p_{t-1}(e,e)+p_{t+1}(e,e))\).

The first example of a finitely generated group with a heat decay equivalent to \(\exp(-t^{1/3})\) was given by N. Th. Varopoulos. He showed that the heat decay on the wreath product \((\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}\) is equivalent to the expectation \(E[2^{-R_t}]\) where \(R_t\) is the random variable which counts the number of visited sites during time \(t\) for a random walk on \(\mathbb{Z}\) [see Appendix II of Bull. Sci. Math., II. Sér. 107, 337-344 (1983; Zbl 0532.60009) and the correction ibid. 108, 263-273 (1984; Zbl 0546.60008)]. The heat decay of the standard wreath product \(K\wr Q\) of two finitely generated groups \(K\) and \(Q\) with \(Q\) infinite (otherwise, the Cartesian product of \(|Q|\) copies of \(K\) is a subgroup of index \(|Q|\) in \(K\wr Q\)) behaves like \(\exp(-t^{1/3})\) if and only if \(K\) is a finite non-trivial group and \(Q\) is a finite extension of \(\mathbb{Z}\). This follows from technics developed by the authors [in Ann. Probab. 30, No. 2, 948-977 (2002; Zbl 1021.60004)].

G. Alexopoulos established the lower bound \(p_{2t}(e,e)\succsim\exp(-t^{1/3})\) for polycyclic groups [Can. J. Math. 44, No. 5, 897-910 (1992; Zbl 0762.31003)]. The main result of the paper under review is the generalization of this lower bound to the class of finitely generated solvable groups of finite Prüfer rank. Recall that a finitely generated group has Prüfer rank if there is an integer \(r\), such that any of its finitely generated subgroups admits a generating set of cardinality less or equal to \(r\).

Here are the main results of the paper under review.

Theorem 1.1. Let \(\Gamma\) be a finitely generated virtually solvable group of finite Prüfer rank. Then the heat decay in \(\Gamma\) satisfies \(p_{2t}(e,e)\succsim\exp(-t^{1/3})\).

Corollary 1.2. Let \(\Gamma\) be a finitely generated virtually solvable group of finite Prüfer rank. The heat decay of \(\Gamma\) satisfies \(p_{2t}(e,e)\sim\exp(-t^{1/3})\) if and only if \(\Gamma\) is not virtually nilpotent.

Reviewer: Alireza Abdollahi (Isfahan)

### MSC:

20F16 | Solvable groups, supersolvable groups |

60G50 | Sums of independent random variables; random walks |

20F69 | Asymptotic properties of groups |

20P05 | Probabilistic methods in group theory |

20F05 | Generators, relations, and presentations of groups |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

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