Optimal estimates of the number of links of basis horizontal broken lines for 2-step Carnot groups with horizontal distribution of corank 1

Abstract

For a 2-step Carnot group D_n, "dim" D_n=n+1, with horizontal distribution of corank 1, we proved that the minimal number N_(X_(D_n ) ) such that any two points u,v∈D_n can be joined by some basis horizontal k-broken line (i.e. a broken line consisting of k links) L_k^(X_(D_n ) ) (u,v), k≤N_(X_(D_n ) ), does not exeed n+2. The examples of D_n such that N_(X_(D_n ) )=n+i, i=1,2. were found. Here X_(D_n )={X_1,…,X_n} is the set of left invariant basis horizontal vector fields of the Lie algebra of the group D_n, and every link of L_k^(X_(D_n ) ) (u,v) has the form "exp"(asX_i)(w), s∈[0,s_0], a=const.