Divergence function of the braided Thompson group (Q6046574)

Let \(G\) be a finitely generated group with a finite generating set \(S\). For any \(\delta \in (0,1)\), the \(\delta\)-divergence function of the Cayley graph \(\mathrm{Cay}(G,S)\) of \(G\) is the smallest function \(f_{\delta} : \mathbb{R}^{+} \to \mathbb{R}^{+}\) such that every two vertices of \(\mathrm{Cay}(G,S)\) at distance \(x\) from the identity can be connected by a path of length at most \(f_{\delta}(x)\) while avoiding the ball of radius \(\delta x\) with center at the identity. We say that the group \(G\) has a linear divergence function if there exists \(\delta \in (0, 1)\) such that the \(\delta\)-divergence function of \(G\) is equivalent to a linear function. For \(n \ge 3\), the Artin braid groups \(B_n\) have linear divergence functions. \textit{G. Golan} and \textit{M. Sapir} [Geom. Dedicata 201, 227--242 (2019; Zbl 1453.20055)] proved that the Thompson groups \(F\), \(T\) and \(V\) have linear divergence functions. In this paper, the author proves that the braided Thompson group \(BV\) has a linear divergence function using a similar method of proof as in [loc. cit.]. By [\textit{C. Druţu} et al., Trans. Am. Math. Soc. 362, No. 5, 2451--2505 (2010; Zbl 1260.20065)], a group \(G\) has a linear divergence function if and only if none of the asymptotic cones of \(G\) has a cut point. Hence, the above result implies that none of the asymptotic cones of \(BV\) has a cut point.