A remark on complete non-expanding Ricci solitons (Q998813)

In a recent note [Proc. Am. Math. Soc. 136, No.~5, 1803--1806 (2008; Zbl 1152.53057)], \textit{W. Wylie} proved that a complete shrinking Ricci soliton has finite fundamental group. In the present note, the author shows that if \(M\) is a complete non-expanding Ricci soliton, then under a technical assumption on the universal covering of \(M\), any finitely generated subgroup of \(M\) has polynomial growth. Let \((M,g)\) be an \(n\)-dimensional Riemannian manifold and Ric be the Ricci tensor. In this paper the author is interested in studying complete Riemannian manifolds \((M,g)\) with a vector field \(X\) such that, for some real constant \(\lambda\), \[ \text{Ric}+\mathcal{L}_Xg\geq\lambda g,\tag{1.1} \] where \(\mathcal{L}_Xg\) is the Lie derivative of \(g\) with respect to the vector field \(X\). If Ric+\(\mathcal{L}_Xg=\lambda g\), the metric is called a complete Ricci soliton. A complete Ricci soliton is said to be shrinking, steady or expanding if \(\lambda>0,\lambda=0\) or \(\lambda<0\), respectively. For \(\lambda>0\), \textit{M. Fernádez-López} and \textit{E. Garcia-Rio} [Math. Ann. 340, No.~4, 893--896 (2008; Zbl 1132.53023)] showed that \((M,g)\) is compact if and only if the \(L^\infty\) norm of \(X\) is bounded. Moreover, in such a case, it has a finite fundamental group. The assumption of bounded \(\|X\|_\infty\) is necessary to show \(M\) that is compact since, for example, the Euclidean space with the vector field \(X(v)=v\) for all \(v\in \mathbb{R}^n\), satisfies (1.1). However Wylie showed (op.cit.) that the fundamental group must be finite even in the noncompact case. The author is interested in studying complete non-expanding Ricci solitons. More generally, he studies \((M,g)\) satisfying (1.1) with \(\lambda\) being nonnegative which may not be constant. Let \(H\) be a finitely generated subgroup of a group \(G\), and \(\{g_1,g_2,\dots,g_p\}\) be a set of generators of \(H\). The \textit{growth function} \(\phi\) of \(H\) associated with \(\{g_1,g_2,\dots,g_p\}\) is defined as follows: For each positive integer \(s\) let \(\phi(s)\) be the number of distinct group elements in \(G\) which can be expressed as words of length not greater than \(s\) in \(\{g_1,g_2,\dots,g_p\}\) and their inverses. \(H\) is said to have polynomial growth if for some constants \(c\) and \(m\), \(\phi(s)\leq c.s^m\) for all \(s\in \mathbb{Z}^+\). Let \(\widetilde{M}\) be the universal covering of \(M\). Then the metric on \(M\) induced a metric on \(\widetilde{M}\) which also satisfies (1.1) with a new vector field \(\widetilde{X}\) being the pullback of the original vector field \(X\). Let \(p\) be a fixed point in \(\widetilde{M}\) and \(M(r)=max\{|\widetilde{X}(x)|:d(x,p)\leq r\}\). The author proves the following \textbf{Theorem 1.1}. If \(M\) satisfies (1.1) with \(\lambda\geq 0\) such that \(M(r)\leq C/r\) for some constant \(C\), then any finitely generated subgroup \(H\) of the fundamental group \(\pi_1(M)\) has polynomial growth. More precisely, the growth function of \(H\) satisfies \(\phi(s)\leq c.s^n\), where \(n\) is the dimension of \(M\). The techniques used by the author in the proof of Theorem 1.1 is a modified of one used by \textit{J. W. Milnor} in [J. Differ. Geom. 2, 1--7 (1968; Zbl 0162.25401)].