Divergence functions of higher-dimensional Thompson's groups (Q6656371)

Thee three Thompson's groups \(V < T < V\), introduced by \textit{R. Thompson} [unpublished (\(\sim\)1965)], have a collection of unusual properties. All three Thompson groups are infinite but finitely presented. The groups \(V\) and \(T\) are examples of infinite but finitely-presented simple groups, the group \(F\) is not simple but its derived subgroup \([F,F]\) is. Higher-dimensional Thompson's groups, denoted by \(nV\) (\(n \in \mathbb{N}\), \(n\geq 2\)) are some such groups defined by \textit{M. G. Brin} in [Geom. Dedicata 108, 163--192 (2004; Zbl 1136.20025)]. The group \(V\) acts on the Cantor set \(\mathfrak{C}\), and the group \(nV\) acts on the powers of the Cantor set \(\mathfrak{C}^{n}\). \textit{G. Golan} and \textit{M. Sapir}, in [Geom. Dedicata 201, 227--242 (2019; Zbl 1453.20055)], proved that \(V\), \(T\) and \(F\) have linear divergence functions.\N\NIn the paper under review, the author proves Theorem 1.1: Higher-dimensional Thompson's groups have linear divergence functions. By \textit{C. Druţu} et al. [Trans. Am. Math. Soc. 362, No. 5, 2451--2505 (2010; Zbl 1260.20065); corrigendum ibid. 370, No. 1, 749--754 (2018; Zbl 1375.20047)], Theorem 1.1 implies none of the asymptotic cones of \(nV\) has a cut-point.