The authors study quasimeromorphic mappings on Carnot groups. Quasimeromorphic mappings \(f: {\mathbb R}^n \to \overline{{\mathbb R}}^n\) generalize quasiregular mappings in the same way as meromorphic functions generalize analytic function [see e.g. S. Rickman’s monograph “Quasiregular mappings” (1993; Zbl 0816.30017)]. A Carnot group \(G\) is a simply connected Lie group \(G\) whose Lie algebra decomposes into the direct sum of vector spaces \(V_1,\dots,V_m\) satisfying the relations \([V_1,V_k]=V_{k+1}\) for \(1\leq k<m\) and \([V_1,V_m]=\{0\}\).
The fundamental difference between the situation on a Carnot group and on a Euclidean space is that, in general, on a Carnot group no inversions may be defined, and hence, conformal mappings may be very few. Some results are given on so-called polarizable Carnot groups [Z. M. Balogh and J. T. Tyson, Math. Z. 241, No. 4, 697–730 (2002; Zbl 1015.22005)]. The main advantage of polarizable Carnot groups is that they admit an analogue of a polar coordinate system.
The proof of the following result is outlined. Suppose that \(G\) is a polarizable Carnot group and \(f: G\to \overline{G}\) is a nonconstant \(K\)-quasimeromorphic mapping. Then there exists a set \(E\subset [1,\infty)\) and a constant \(C(Q,K)<\infty\) such that \[ \lim_{r\to\infty}\sup_{r\notin E}\sum_{j=0}^q \bigg(1-{{n(r,a_j)}\over{v(r,1)}}\bigg)_+ \leq C(Q,K)\;\;\text{ and }\;\;\int_E{{dr}\over{r}}<\infty, \] for any distinct points \(a_0,a_1,\dots,a_q\) in \(\overline{G}\). Here the quantity \(Q=\sum_{i=1}^m i \dim V_i\) is called the homogeneous dimension of \(G\), \[ n(r,y) =\sum_{x\in f^{-1}(y)\cap B(0,r)}i(x,f), \] \(i(x,f)\) is the local (topological) index of \(f\) at \(x\) and \(v(r,s)\) denotes the mean value of the function \(n(r,y)\) on the sphere \(S(0,s)\).