Divergence functions of higher-dimensionalThompson’s groups

We prove that higher-dimensional Thompson's groups have linear divergence functions.By the work of Drut , u, Mozes, and Sapir, this implies none of the asymptotic cones of nV has a cut-point. IntroductionThompson's groups F, T , and V are finitely presented infinite groups defined by Richard Thompson in the 1960s.They are all known to be mysterious groups.For example, T and V are the first examples of finitely presented, infinite, and simple, and it is known that the amenability of F is a difficult open problem.Because they have several unpredictable properties, by focusing on such properties, many "generalized" Thompson's groups were also defined.Higher-dimensional Thompson's groups, denoted by 2V, 3V, . . ., are some such groups defined by Brin [3].The group V acts on the Cantor set C, and the group nV acts on the powers of the Cantor set C n .It is known that nV is also finitely presented [4; 14] and simple [3; 5].In addition, it was shown that for n, m >0 , the group nV is isomorphic to mV if and only if n = m holds [1].In [16], it was proved that nV has Serre's property FA, and hence is one-ended.In 2018, Golan and Sapir showed that the divergence functions of F, T , and V are linear [12].This function was first mentioned by Gromov [13], and later, Gersten gave the formal definition [11] as a quasi-isometric invariant of geodesic metric spaces.Roughly speaking, the order of the function indicates whether the Cayley graphs of the group are "close" to the Euclidean or hyperbolic spaces.In fact, the orders of the functions of the direct powers of the infinite cyclic group 2 , 3 , . . .are linear, and it is known that the orders of the functions of hyperbolic groups are at least exponential [2].In [12], they asked whether their proof could be extended to generalized Thompson's groups.In recent years, similar results have been obtained for some groups by extending the original arguments [17; 19; 18].For recent results on functions of groups other than generalized Thompson's groups, see [15].