This contains the correct lower estimate for the central value \(\mu*\dots*\mu(e)\) of the \(n\)th convolution of a symmetric probability measure \(\mu\) on certain discrete polycyclic groups \(G\) of exponential volume growth whose support is finite and degenerates \(G\) [cf. M. S. Raghunathan, Discrete subgroups of Lie groups (1972; Zbl 0254.22005)]: namely, \(\exists A\), \(a>0\) such that \(\mu^{*n}(e)\geq A\exp(- an^{1/3})\). Varopoulos had shown that \(\exists B\), \(b>0\) such tht \(\mu^{*n}(e)\leq B\exp(-bn^{1/3})\) for any discrete group of exponential volume growth. For the case of the heat equation in the context of a connected, unimodular, amenable Lie group of exponential volume growth studied by Varopoulos and co-workers in recent years, the author proves that for the heat kernel there exist \(A\), \(a>0\) such that \(p_ t(x,x)\geq A\exp(-at^{1/3})\), \(\forall x\in C\), \(t\geq 1\). In view of the similar upper bound established by N. Th. Varopoulos [Proc. Int. Congr. Math. Kyoto/Japan 1990, Vol. II, 951-957 (1991; Zbl 0744.43006)], this gives the asymptotic behaviour of the central value \(p_ t(x,x)\) as \(t\to+\infty\). The arduous proofs involve properties of the entropy of random walks on groups due to several writers including the author.