Stable exponents in discrete groups. (Q1863078)

Let \(G\) be an arbitrary group. The stable exponent \(p_{+}(g)\) of an element \(g \in G\) is defined as \[ p_{+}(g)=\mathop{\lim \sup}_{n \to \infty} \frac{p(g^n)}{n}, \] where \(p(g^n)\) denotes the supremum of the set of integers \(k \geq 1\) such that \(g^n\) admits a \(k\)th root in \(G\). \textit{M.Gromov} [In: Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 75-263 (1987; Zbl 0634.20015)] has observed that in a word hyperbolic group the stable exponent of every non-torsion element is an integer. The author of the paper proves that this is also true for non-torsion element in finitely generated nilpotent groups. Besides this result, the author shows that, for any rational number \(r \geq 1\), there exists a torsion-free CAT(0) group containing an element whose stable exponent is equal to \(r\).