\n\n
\n<\/p>\n
\n\n\n\n\n\n\n$}$<\/p>\n
Milnor\u5173\u4e8e\u6709\u9650\u751f\u6210\u7fa4\u7684\u5206\u7c7b\u5b9a\u7406\u7684\u4e00\u4e2a\u51e0\u4f55\u65b9\u6cd5<\/span>\n\u620e\u5c0f\u6625<\/span>\n06\/27\/2019<\/span><\/p>\n \u4ee4 \u5e42\u96f6\u7fa4\u53ef\u770b\u4f5c\u662f“\u5dee\u4e0d\u591a\u7684\u4ea4\u6362\u7fa4”, \u5e42\u96f6\u7fa4\u90fd\u662f\u53ef\u89e3\u7fa4. \u5e42\u96f6\u7fa4\u7684\u5b9a\u4e49\u5982\u4e0b. <\/span> \u5229\u7528[3<\/a>], \u6211\u4eec\u77e5\u9053, \u5b58\u5728\u9ece\u66fc\u4e07\u6709\u8986\u76d6\u7684\u7b49\u53d8\u6536\u655b\u5b50\u5e8f\u5217$(\\tilde{M}_i,\\tilde{p}_i,\\Gamma_i)\\GHto(\\tilde{X}, \\tilde{p}, G)$, \u800c\u4e14\u5b58\u5728$\\epsilon>0$, \u4f7f\u5f97\u5b50\u7fa4 Chen–Rong–Xu \u4e3b\u8981\u9700\u8981\u8bc1\u660e\u5982\u4e0b\u7684\u5b9a\u7406, \u5b83\u7ed9\u51fa\u4e86\u5224\u65ad$\\Gamma$\u662f\u81f3\u591a\u591a\u9879\u5f0f\u589e\u957f\u7684\u4e00\u4e2a\u5145\u5206\u6761\u4ef6. \u5176\u8bc1\u660e\u548c[10<\/a>]\u7c7b\u4f3c.
\n\n
\n<\/div>
\n1. \u5b57\u957f\u71b5<\/a><\/span>\n\n\u6709\u9650\u5355\u7fa4\u7684\u5206\u7c7b\u88ab\u5b8c\u5168\u89e3\u51b3, \u800c\u65e0\u9650\u7fa4\u4e2d, \u5173\u4e8eAbel\u7fa4\u7684\u5206\u7c7b\u4e5f\u5df2\u7ecf\u6e05\u695a. \u4e00\u4e2a\u81ea\u7136\u7684\u95ee\u9898\u662f\u6bd4Abel\u7fa4\u7a0d\u5fae\u590d\u6742\u70b9\u7684\u7fa4\u7684\u5206\u7c7b\u95ee\u9898. \u800c\u4e3a\u4e86\u523b\u753b\u4e00\u4e2a\u65e0\u9650\u7fa4\u7684\u590d\u6742\u7a0b\u5ea6, \u53ef\u4ee5\u7528\u5982\u4e0b\u7684\u5b57\u957f\u71b5<\/span>\u6765\u5b9a\u4e49.
\n<\/span>
\n\\[
\n\\lvert \\gamma \\rvert_S=\\min\\left\\{ k: \\gamma=\\gamma_{i_1}\\gamma_{i_2}\\cdots\\gamma_{i_k}, \\gamma_{i_j}\\in S \\right\\}.
\n\\]<\/p>\n
\n\\[
\n\\lvert \\Gamma(R,S) \\rvert={}^\\sharp\\left\\{ \\gamma\\in \\Gamma: \\lvert \\gamma \\rvert_S\\leq R \\right\\},
\n\\]
\n\u5373$\\Gamma$\u4e2d\u5b57\u957f\u5c0f\u4e8e$R$\u7684\u5143\u7d20\u4e2a\u6570. \u5b9a\u4e49$(\\Gamma,S)$\u7684\u5b57\u957f\u71b5<\/span>\u4e3a
\n\\[
\nh_w(\\Gamma,S)\\mathpunct{:}=\\lim_{R\\to\\infty}\\frac{\\ln\\lvert \\Gamma(R,S) \\rvert}{R}.
\n\\]
\n<\/div>
\n
\n\u9996\u5148, \u6211\u4eec\u53ef\u4ee5\u8bc1\u660e
\n<\/span>
\n<\/div>
\n
\n\\[
\nd_w(\\gamma_1,\\gamma_2)=\\lvert \\gamma_1\\gamma_2^{-1} \\rvert_S.
\n\\]
\n
\n\u4e3a\u4e86\u533a\u5206\u6709\u9650\u751f\u6210\u7684(\u65e0\u9650)\u7fa4\u7684\u590d\u6742\u5ea6, \u6211\u4eec\u5f15\u8fdb\u4e24\u4e2a\u6982\u5ff5.
\n<\/span>
\n\\[
\n\\lvert \\Gamma(R,S) \\rvert\\leq R^m.
\n\\]
\n<\/div>
\n<\/span>
\n<\/div>
\n2. Milnor\u95ee\u9898<\/a><\/span>\n\n\u660e\u663e, \u81f3\u591a\u591a\u9879\u5f0f\u589e\u957f\u7684\u6709\u9650\u751f\u6210\u7fa4\u7684\u5b57\u957f\u71b5\u7b49\u4e8e0. \u800c\u53cd\u8fc7\u6765\u5c31\u662f\u8457\u540d\u7684Milnor\u95ee\u9898.
\n<\/span>
\n<\/div>
\n\u540e\u6765, Grigorchuk [4<\/a>] \u901a\u8fc7\u6784\u9020\u4e00\u4e2a\u6709\u9650\u751f\u6210\u4f46\u4e0d\u80fd\u6709\u9650\u8868\u793a\u7684\u65e0\u9650\u7fa4\u7ed9\u51fa\u4e86Milnor\u95ee\u9898\u7684\u4e00\u4e2a\u5426\u5b9a\u56de\u7b54. \u8fdb\u800cMilnor\u95ee\u9898\u5e94\u8be5\u6539\u4e3a
\n<\/span>
\n<\/div>
\n\u6ce8\u610f, \u8fd9\u91cc\u6709\u9650\u8868\u793a\u5bf9\u51e0\u4f55\u5b66\u5bb6\u7684\u610f\u4e49\u91cd\u5927, \u56e0\u4e3a\u8fd9\u6837$\\Gamma$\u53ef\u4ee5\u89c6\u4e3a\u4e00\u4e2a\u7d27\u81f4$n$($n\\geq4$)-\u7ef4\u9ece\u66fc\u6d41\u5f62\u7684\u57fa\u672c\u7fa4. \u53c2\u8003[2<\/a>,Thm.~5.1.1]. \u800c\u5bf9\u8fd9\u79cd\u6d41\u5f62\u51e0\u4f55\u5b66\u5bb6\u6709\u6df1\u5165\u7684\u7814\u7a76.
\n3. \u4e00\u4e2a\u51e0\u4f55\u7684\u7b49\u4ef7\u523b\u753b<\/a><\/span>\n\n\u6211\u4eec\u5173\u4e8e\u7b49\u4ef7\u523b\u753b\u7684\u542f\u53d1\u5728\u4e8e\u5982\u4e0b\u7684Gromov\u7684\u4e00\u4e2a\u731c\u60f3.
\n3.1. Gromov\u7684\u4e00\u4e2a\u731c\u60f3<\/a><\/span>\n\n\u6211\u4eec\u77e5\u9053, \u5bf9\u4efb\u4f55\u4e00\u4e2a\u7fa4$N$, \u5176$1$\u9636\u4ea4\u6362\u5b50\u7fa4\u5b9a\u4e49\u4e3a$N_1=[N,N]=\\left\\{ ghg^{-1}h^{-1}: g,h\\in N \\right\\}$; \u9012\u5f52\u5730\u5b9a\u4e49\u9ad8\u9636\u4ea4\u6362\u5b50\u7fa4$N_k=[N_{k-1},N_{k-1}]$, $N_0=N$. \u5982\u679c\u5b58\u5728$n$\u4f7f\u5f97$N_n=e$, \u5219$N$\u79f0\u4e3a\u53ef\u89e3\u7fa4<\/span>. \u4ea4\u6362\u5b50\u7fa4\u523b\u753b\u4e86\u4e00\u4e2a\u7fa4\u7684\u4ea4\u6362\u6027. \u660e\u663e, \u4ea4\u6362\u7fa4\u7684\u4ea4\u6362\u5b50\u7fa4\u662f\u5e73\u51e1\u7684. \u5bb9\u6613\u8bc1\u660e\u5546\u7fa4
\n\\[
\nN\/N_1
\n\\]
\n\u662f\u4e00\u4e2a\u4ea4\u6362\u7fa4, \u79f0\u4e3a$N$\u7684\u963f\u8d1d\u5c14\u5316\u5b50\u7fa4<\/span>. \u4e8b\u5b9e\u4e0a, \u53ef\u4ee5\u8bc1\u660e\u4ea4\u6362\u5b50\u7fa4\u662f\u4f7f\u5f97$N\/N_1$\u4e3a\u4ea4\u6362\u7fa4\u7684\u6700\u5c0f\u5b50\u7fa4.<\/p>\n
\n<\/span>
\n\\[
\nN=N_0\\triangleright N_1\\triangleright\\cdots\\triangleright N_s=e,\\quad N_{i+1}=[N,N_i].
\n\\]
\n<\/div>
\n\u7279\u522b\u5730, \u5982\u679c$N$\u662f\u4e00\u4e2a\u6709\u9650\u751f\u6210\u7684\u5e42\u96f6\u7fa4, \u5219\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5982\u4e0b\u65b9\u5f0f\u6765\u6784\u9020\u57fa\u4e8e\u9012\u964d\u5217\u7684\u5bf9\u79f0\u751f\u6210\u96c6:
\n
\n<\/span>
\n<\/div>
\n\u6bd4\u5e42\u96f6\u7fa4\u7a0d\u5fae\u5dee\u4e00\u70b9\u7684\u7fa4\u662f\u6240\u8c13\u7684\u51e0\u4e4e\u5e42\u96f6\u7fa4<\/span>(almost nilpotent group). \u5176\u5b9a\u4e49\u5982\u4e0b
\n<\/span>
\n<\/div>
\n\u4e8b\u5b9e\u4e0a[1<\/a>]*{Lem.~1.4}\u7684\u4e00\u4e2a\u7ed3\u679c\u8bf4\u660e\u51e0\u4e4e\u5e42\u96f6\u7fa4\u4e5f\u662f\u81f3\u591a\u591a\u9879\u5f0f\u589e\u957f\u7684.<\/p>\n
\n\\[
\n\\mathrm{Ric}_M\\geq-(n-1),\\quad \\mathrm{diam}(M)\\leq \\epsilon(n),
\n\\]
\n\u90a3\u4e48\u57fa\u672c\u7fa4$\\pi_1(M)$\u662f\u51e0\u4e4e\u5e42\u96f6\u7684, \u5373\u5b58\u5728\u5e42\u96f6\u5b50\u7fa4$N$\u4f7f\u5f97\u5176\u6307\u6570\u6ee1\u8db3
\n\\[
\n[\\pi_1(M):N]\\leq C(n).
\n\\]
\n<\/div>
\n\u4e00\u4e2a\u81ea\u7136\u7684\u95ee\u9898\u662f, \u4e0a\u8ff0\u6761\u4ef6\u4e2d$\\epsilon(n)$\u7684\u4e0a\u786e\u754c\u662f\u591a\u5c11? \u53d6\u5f97\u4e0a\u786e\u754c\u65f6\u7684\u6d41\u5f62\u662f\u54ea\u4e9b? Chen-Rong-Xu [1<\/a>]\u53d1\u73b0\u5728\u4e00\u4e2a\u81ea\u7136\u6761\u4ef6\u4e0b, \u8fd9\u4e00\u95ee\u9898\u548cMilnor\u731c\u60f3\u7b49\u4ef7. \u4e3a\u6b64, \u6211\u4eec\u8fd8\u9700\u8981\u5b9a\u4e49\u4e00\u79cd\u71b5.
\n<\/span>
\n\\[
\nh(M) \\mathpunct{:}=\\lim_{R\\to\\infty}\\frac{\\ln\\lvert B_R(\\tilde{p}) \\rvert}{R},\\quad\\tilde{p}\\in\\tilde{M},
\n\\]
\n\u4e3a\u6d41\u5f62$M$\u7684\u4f53\u79ef\u71b5(volume entropy).
\n<\/div>
\n\u53ef\u4ee5\u8bc1\u660e(\u53c2\u8003[6<\/a>]), \u4f53\u79ef\u71b5\u603b\u662f\u5b58\u5728\u7684, \u800c\u4e14\u4e0e$\\tilde{p}\\in\\tilde{M}$\u7684\u9009\u62e9\u65e0\u5173.
\n
\n<\/div>
\n<\/span>
\n\\[
\n\\mathrm{Ric}_M\\geq-(n-1),\\quad d\\geq \\mathrm{diam}(M), \\quad h(M)<\\epsilon(n,d),
\n\\]
\n\u90a3\u4e48$\\pi_1(M)$\u6709\u4e00\u4e2a\u5177\u6709\u6709\u9650\u6307\u6570\u7684\u5e42\u96f6\u5b50\u7fa4, \u5373$\\pi_1(M)$\u662f\u51e0\u4e4e\u5e42\u96f6\u7684.
\n<\/div>
\nChen–Rong–Xu[1<\/a>]\u7684\u4e3b\u8981\u5b9a\u7406\u5982\u4e0b
\n<\/span>
\n<\/div>
\n
\n\\[
\nh_w(\\pi_1(M),S)\\leq h(M)\\leq C h_w(\\pi_1(M),S),
\n\\]
\n\u5373$h_w(\\pi_1(M),S)=0\\iff h(M)=0$. \u8fd9\u91cc$C$\u662f\u4e00\u4e2a\u4f9d\u8d56\u4e8e$S$\u7684\u5e38\u6570.
\n<\/div>
\n\u4ece\u800c\u7ed3\u5408\u4efb\u4f55\u6709\u9650\u8868\u793a\u7684\u6709\u9650\u751f\u6210\u7fa4\u90fd\u53ef\u4ee5\u89c6\u4e3a\u67d0\u4e2a$n\\geq4$\u7684\u7d27\u6d41\u5f62$M$\u7684\u57fa\u672c\u7fa4$\\Gamma=\\pi_1(M)$, \u800c\u4e14\u6b64\u65f6$h_w(\\pi_1(M),S)=0\\iff h(M)=0$. \u4ece\u800cChen–Rong–Xu\u7684\u731c\u60f3\u81ea\u7136\u8574\u542bMilnor\u95ee\u9898\u7684\u80af\u5b9a\u7b54\u6848. \u800cChen–Rong–Xu\u7684\u4e3b\u8981\u5de5\u4f5c\u662f\u8bc1\u660e$h(M)\\ll1$\u65f6, \u6709$\\pi_1(M)$\u662f\u51e0\u4e4e\u5e42\u96f6\u7684.
\n4. \u8bc1\u660e\u7684\u57fa\u672c\u60f3\u6cd5<\/a><\/span>\n\n\u7531Gromov\u7d27\u6027\u5b9a\u7406, \u7b49\u4ef7\u4e8e\u8bc1\u660e\u5728Milnor\u95ee\u98982<\/a>\u6709\u80af\u5b9a\u7b54\u6848\u65f6, \u8bc1\u660e\u5bf9\u4e00\u4e2a$n$-\u7ef4\u7d27\u6d41\u5f62\u7684\u6536\u655b\u5e8f\u5217$M_i\\GHto X$, \u5176\u4e2d$M_i$\u6ee1\u8db3
\n\\[
\n\\mathrm{Ric}_{M_i}\\geq -(n-1),\\quad d\\geq \\mathrm{diam}(M_i),\\quad h(M_i)\\to 0,
\n\\]
\n\u6210\u7acb\u5bf9\u5145\u5206\u5927\u7684$i$, \u6709$\\Gamma_i=\\pi_1(M_i)$\u662f\u51e0\u4e4e\u5e42\u96f6\u7684.<\/p>\n
\n\\[
\n\\Gamma_{i,\\epsilon}=\\left\\{ \\gamma_i\\in\\Gamma_i: d(\\tilde{q}_i,\\gamma_i(\\tilde{q}_i))<\\epsilon,\\,\\tilde{q}_i\\in B_1(\\tilde{p}_i) \\right\\}
\n\\]
\n\u65f6\u6b63\u89c4\u7684. \u6b64\u5916, \u5bf9\u5145\u5206\u5927\u7684$i$,
\n\\[
\n\\Gamma_{i}\/\\Gamma_{i,\\epsilon}\\simeq G\/G_0
\n\\]
\n\u8fd9\u91cc, $G_0$\u65f6$G$\u7684\u5355\u4f4d\u5143\u5206\u652f. \u73b0\u5728, \u6211\u4eec\u5206\u5982\u4e0b\u4e24\u79cd\u60c5\u51b5:
\n
\n\\begin{equation}\\label{eq:Nilpotent-condition}
\n1\\to N_i\\to \\Gamma_i\\xrightarrow{\\pi_i}\\to\\Lambda_i\\to1,\\quad
\nN_i=N_{i1}\\triangleright\\cdots N_{ik}\\triangleright N_{ik+1}=\\mathrm{Tor}(N_i),
\n\\end{equation}
\n\u6ee1\u8db3
\n<\/span>
\n
\n<\/div>
\n\u56e0\u6b64, \u6211\u4eec\u53ea\u9700\u9a8c\u8bc1\u5b9a\u740611<\/a>\u4e2d\u7684\u4e24\u4e2a\u6761\u4ef6. \u6ce8\u610f\u5230A\u53ef\u7531
\n\\[
\nh_w(\\Lambda,\\bar{S}_0)\\leq h_w(\\Gamma_i,S_i)\\to0,
\n\\]
\n\u4ee5\u53caMilnor\u95ee\u98982<\/a>\u7684\u80af\u5b9a\u7b54\u6848\u5f97\u5230. \u800cA\u7684\u9a8c\u8bc1\u57fa\u4e8e$h_w(\\Gamma)$\u7684\u4e00\u4e2a\u4e0b\u754c\u4f30\u8ba1, \u5176\u4e2d$\\Gamma$\u6ee1\u8db3
\n\\[
\n1\\to \\mathbb{Z}^k\\to\\Gamma\\to \\mathbb{Z}\\to 1,
\n\\]
\n\u53c2\u8003[9<\/a>], \u4ee5\u53ca$\\bar{\\rho}_h$\u7684\u6709\u9650\u6027(\u89c1[1<\/a>]*{Lem.~2.1}).
\n<\/ol>\n<\/p>\n