Growth of fundamental groups and isoembolic volume and diameter (Q2758998)

Let \(M:= M^n\) denote a compact Riemannian \(n\)-manifold without boundary, and let \(\pi_1:= \pi_1(M, p)\), \(d(M)\), \(v(M)\) and \(i(M)\) denote ist fundamental group, diameter, volume and injectivity radius, respectively. For finitely generated \(\pi_1\), let \(mg(\pi_1)\) denote the minimal number of generators of \(\pi_1\). The entropy \(h(\pi_1)\) is measure by the number of distinct words up to a given length in terms of generators and their inverses of \(\pi_1\), the entropy \(h(M)\) of \(M\) is characterized by the volume growth of its universal cover. The isoembolic volume \(V_e(M):= v(M)/(i(M))^n\) and the isoembolic diameter \(d_e(M):= d(M)/i(M)\) are defined. The author's estimates are summarized in Theorem 2: NEWLINE\[NEWLINEmg(\pi_1)\leq [cV_e(M)]^{14d_e(M)}\leq [cV_e(M)]^{4cV_e(M)};\;h(\pi_1)\leq 14d_e(M)^.\log cV_e(M);NEWLINE\]NEWLINE where \(c\) is a constant depending only on the dimension \(n\). This is a consequence of the more technical Theorem 1, where the packing number \(N(a,c,M)\), i.e. the largest number of disjoint open metric balls of radius \(c\) that can be fitted into any metric ball of radius \(a\) in \(M\), playes the crucial role in case \(6c< a\).