{"id":1810,"date":"2022-05-29T22:43:14","date_gmt":"2022-05-29T14:43:14","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1810"},"modified":"2025-02-26T11:03:07","modified_gmt":"2025-02-26T03:03:07","slug":"free_group_action_proper_maps_and_proper_discontinuity","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/05\/29\/free_group_action_proper_maps_and_proper_discontinuity\/","title":{"rendered":"\u81ea\u7531\u7fa4\u4f5c\u7528, Proper Maps\u4e0e\u7eaf\u4e0d\u8fde\u7eed"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u672c\u5468\u5b85\u5bb6, \u60f3\u7740\u5c3d\u91cf\u628a\u7fa4\u4f5c\u7528\u8fd9\u4e00\u8282\u5185\u5bb9\u7ed9\u5b8c\u7ed3\u4e86~<\/p>\n<p><!--more--><\/p>\n\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/05\/prop-disc.pdf\">prop-disc<\/a><br \/>\n2. <a href=\"https:\/\/mathoverflow.net\/questions\/55726\/properly-discontinuous-action\">Properly Discontinuous Action<\/a><\/p>\n<h3>1. \u81ea\u7531\u7fa4\u4f5c\u7528<\/h3>\n<p>\u5173\u4e8e\u81ea\u7531\u7fa4\u4f5c\u7528\u7684\u5b9a\u4e49, \u4e0a\u6587\u5df2\u7ecf\u4ecb\u7ecd\u8fc7\u4e86, \u6b64\u5904\u4e0d\u518d\u8d58\u8ff0. \u81ea\u7531\u662f\u4e00\u4e2a\u6bd4\u8f83\u62bd\u8c61\u7684\u6982\u5ff5, \u5f15\u8fdb\u81ea\u7531\u8fd9\u4e2a\u6982\u5ff5\u5176\u5b9e\u662f\u4e3a\u4e86\u523b\u753b\u8f68\u9053\u7684\u5300\u9f50\u6027: \u5982\u679c\u4e0d\u81ea\u7531\u5219\u542b\u4e0d\u52a8\u70b9\u7684\u8f68\u9053\u6bd4\u5176\u5b83\u8f68\u9053\u6240\u542b\u7684\u70b9\u6570\u8981\u5c11\u4e9b, \u8fd9\u6837\u4e0d\u540c\u7684\u8f68\u9053\u5c31\u5fc5\u987b\u8981\u533a\u522b\u5bf9\u5f85\u4e86. \u4ece\u76f4\u89c2\u4e0a\u6765\u7406\u89e3, \u82e5\u8ba8\u8bba\u7684\u7fa4\u4f5c\u7528\u662f\u4e0d\u81ea\u7531\u7684, \u5728\u4e00\u6761\u542b\u6709\u4e0d\u52a8\u70b9$x_0$\u7684\u8f68\u9053\u4e2d, \u7531\u4e8e\u6ee1\u8db3$g x_0 = x_0$\u7684$g \\in G$\u4e0d\u4ec5\u4ec5\u53ea\u6709\u7fa4$G$\u4e2d\u7684\u5355\u4f4d\u5143$e$, \u5219\u6709\u591a\u4e2a\u7fa4\u5143\u7d20$g$\u5c06\u70b9$x_0$\u4f9d\u65e7\u53d8\u6362\u5230\u70b9$x_0$, \u4ece\u800c\u70b9$x_0$\u6240\u5728\u7684\u8f68\u9053\u542b\u6709\u7684\u70b9\u7684\u6570\u91cf\u6bd4\u5176\u5b83\u8f68\u9053\u6240\u542b\u7684\u70b9\u6570\u8981\u5c11\u4e9b.<\/p>\n<h3>2. Proper Maps<\/h3>\n<p>\u5173\u4e8eProper Maps\u7684\u5b9a\u4e49, \u5176\u5b9e\u662f\u6709\u5f88\u591a\u7248\u672c\u7684, \u76ee\u524d\u6700\u5e38\u7528\u7684\u5b9a\u4e49\u662f: \u82e5\u5bf9\u4e8e\u4efb\u610f\u7d27\u96c6$K \\subset X$\u4e0e\u8fde\u7eed\u6620\u5c04$f: X \\to Y$, $f^{-1}(K)$\u4ea6\u4e3a\u4e00\u4e2a\u7d27\u96c6, \u5219\u8fde\u7eed\u6620\u5c04$f$\u662f\u4e00\u4e2aProper Map. \u66f4\u8fdb\u4e00\u6b65\u5730, \u82e5\u8fde\u7eed\u6620\u5c04$f: X \\to Y$\u662f\u4e00\u4e2a\u95ed\u6620\u5c04, \u4e14$f^{-1} $$ (y), y \\in Y$\u4ea6\u4e3a\u4e00\u4e2a\u7d27\u96c6, \u5219\u8fde\u7eed\u6620\u5c04$f$\u662fBourbaki-Proper\u7684. \u53ef\u4ee5\u8bc1\u660e,<br \/>\n$\\\\$ a) \u82e5$X$\u662f\u4e00\u4e2aHausdorff\u7a7a\u95f4, $Y$\u662f\u5c40\u90e8\u7d27\u7684\u62d3\u6251\u7a7a\u95f4, \u5219\u8fde\u7eed\u6620\u5c04$f: X \\to Y$\u662fBourbaki-Proper\u7684\u5f53\u4e14\u4ec5\u5f53\u8fde\u7eed\u6620\u5c04$f$\u662f\u4e00\u4e2aProper Map.<br \/>\n$\\\\$ b) \u82e5$X$\u4e0e$Y$\u5747\u4e3a\u7b2c\u4e00\u53ef\u6570\u7684Hausdorff\u7a7a\u95f4, \u5219\u8fde\u7eed\u6620\u5c04$f: X \\to $$ Y$\u662fBourbaki-Proper\u7684\u5f53\u4e14\u4ec5\u5f53\u8fde\u7eed\u6620\u5c04$f$\u662f\u4e00\u4e2aProper Map.<\/p>\n<h3>3. \u7eaf\u4e0d\u8fde\u7eed<\/h3>\n<p>\u5173\u4e8e\u7eaf\u4e0d\u8fde\u7eed\u7684\u5b9a\u4e49, \u4e0eProper Maps\u7684&#8221;\u5904\u5883&#8221; \u7c7b\u4f3c, \u4e5f\u662f\u62e5\u6709\u5f88\u591a\u7248\u672c, \u53c2\u8003\u6750\u65992\u4e2d\u7ed9\u51fa\u4e865\u4e2a\u7248\u672c\u7684\u5b9a\u4e49, \u52a0\u4e0a\u4e66\u4e0a\u63d0\u5230\u7684\u5b9a\u4e49, \u4e00\u5171\u662f6\u4e2a\u7248\u672c\u7684\u5b9a\u4e49. \u53ef\u4ee5\u8bc1\u660e, \u8fd9\u4e9b\u4e0d\u540c\u7248\u672c\u7684\u5b9a\u4e49\u672c\u8d28\u4e0a\u90fd\u662f\u7b49\u4ef7\u7684. \u6b64\u5904\u4ec5\u4ecb\u7ecd\u4e00\u4e0b\u4f7f\u7528Proper Maps\u8bed\u8a00\u5b9a\u4e49\u7eaf\u4e0d\u8fde\u7eed\u7684\u7248\u672c\u4e0e\u4e66\u4e0a\u7684\u5b9a\u4e49.<\/p>\n<p><strong>\u5b9a\u4e491<\/strong> \u7ed9\u5b9a\u79bb\u6563\u7fa4$G$, \u62d3\u6251\u7a7a\u95f4$X$, \u4e0e\u7fa4\u4f5c\u7528$\\alpha: G \\times X \\to X$, \u53ef\u5f97\u81ea\u7136\u6295\u5c04$$\\hat{\\alpha} := \\alpha \\times id_X : G \\times X \\to X \\times X,$$\u5176\u4e2d$id_X : (g, x) \\mapsto x$. \u82e5\u81ea\u7136\u6295\u5c04$\\hat{\\alpha}$\u662f\u4e00\u4e2aProper Map, \u5219\u79f0\u7fa4\u4f5c\u7528$\\alpha$\u662f\u7eaf\u4e0d\u8fde\u7eed\u7684.<\/p>\n<p><strong>\u5b9a\u4e492<\/strong> \u5982\u679c\u4efb\u53d6$x, y \\in X$, \u5b58\u5728$x$\u7684\u90bb\u57df$U_x$\u548c$y$\u7684\u90bb\u57df$U_y$, \u4f7f\u5f97$$\\{ g \\in G | g(U_x) \\cap U_y \\ne \\emptyset \\}$$\u662f\u6709\u9650\u96c6(\u8fd9\u91cc$g(U) = \\{ g(x) | x \\in U \\}$), \u5219\u79f0$q$\u4e3a\u7eaf\u4e0d\u8fde\u7eed\u7fa4\u4f5c\u7528.<\/p>\n<p>\u7eaf\u4e0d\u8fde\u7eed\u4ea6\u662f\u4e00\u4e2a\u6bd4\u8f83\u62bd\u8c61\u7684\u6982\u5ff5, \u5b83\u523b\u753b\u4e86\u6bcf\u4e2a\u8f68\u9053\u4e0a\u7684\u70b9\u7684\u5206\u6563\u7a0b\u5ea6: \u8fd9\u4e9b\u8f68\u9053\u4e0a\u7684\u70b9\u4e0d\u4ec5\u81ea\u5df1\u5f88&#8221;\u5206\u6563&#8221;, \u800c\u4e14\u4e5f\u4e0d\u4f1a&#8221;\u805a\u96c6&#8221; \u5230\u522b\u7684\u8f68\u9053\u4e0a\u7684\u70b9\u9644\u8fd1\u53bb. \u4ece\u76f4\u89c2\u4e0a\u6765\u7406\u89e3, \u8003\u8651\u7eaf\u4e0d\u8fde\u7eed\u5b9a\u4e49\u7684\u5426\u547d\u9898, \u5bf9\u70b9$x \\in X$\u7684\u4efb\u610f\u90bb\u57df$U_x$, \u4e0e\u70b9$y $$ \\in X$\u7684\u4efb\u610f\u90bb\u57df$U_y$, \u4f7f\u5f97$g(U_x) \\cap $$ U_y = \\emptyset$\u6216\u8005$g(U_x) \\cap U_y\\ne \\emptyset$\u6210\u7acb\u7684\u7fa4\u5143\u7d20$g \\in G$\u90fd\u6709\u65e0\u9650\u591a\u4e2a. \u800c\u524d\u4e00\u79cd\u60c5\u51b5\u663e\u7136\u4e0d\u53ef\u80fd\u53d1\u751f, \u6700\u7b80\u5355\u7684\u4e00\u4e2a\u53cd\u4f8b\u4fbf\u662f\u53d6\u70b9$x$\u7684\u90bb\u57df$U_x = X$\u5373\u53ef, \u6b64\u65f6\u4e0d\u53ef\u80fd\u53d6\u4efb\u610f\u7fa4\u5143\u7d20$g \\in G$\u90fd\u4f7f\u5f97$g(X) \\cap U_y $$ = $$ \\emptyset$.<br \/>\n$\\\\$ \u5f53\u8ba8\u8bba\u540c\u4e00\u8f68\u9053\u65f6, \u82e5\u8fd9\u4e00\u6761\u8f68\u9053\u4e0a\u7684\u70b9\u5206\u5e03\u7a20\u5bc6, \u53d6\u8be5\u8f68\u9053\u4e0a\u7684\u4efb\u610f\u4e24\u70b9$x, y$\u4e0e\u70b9$y$\u7684\u4efb\u610f\u90bb\u57df$U_y$, \u5219\u70b9$x$\u7684\u4efb\u610f\u90bb\u57df$U_x$\u6240\u542b\u7684\u5927\u90e8\u5206\u70b9\u4e5f\u662f\u70b9$x$\u6240\u5c5e\u8f68\u9053\u4e0a\u7684\u70b9, \u4e14\u6709\u65e0\u9650\u591a\u4e2a\u7fa4\u5143\u7d20$g \\in G$, \u4f7f\u5f97$g(U_x $$ ) \\cap U_y \\ne \\emptyset$\u6210\u7acb.<br \/>\n$\\\\$ \u5f53\u8ba8\u8bba\u4e0d\u540c\u8f68\u9053$\\mathcal{O}_x, \\mathcal{O}_y$\u65f6, \u5206\u522b\u5728\u8fd9\u4e24\u6761\u8f68\u9053\u4e0a\u53d6\u4e00\u70b9$x, y$, \u7531\u4e8e\u5bf9\u70b9$x$\u7684\u4efb\u610f\u90bb\u57df$U_x$, \u4e0e\u70b9$y$\u7684\u4efb\u610f\u90bb\u57df$U_y$, \u4f7f\u5f97$g(U_x) \\cap U_y \\ne \\emptyset$\u6210\u7acb\u7684\u7fa4\u5143\u7d20$g \\in G$\u6709\u65e0\u9650\u591a\u4e2a, \u4ece\u800c\u8bf4\u660e\u8f68\u9053$\\mathcal{O}_x$\u4e0a\u7684\u70b9\u4f1a&#8221;\u805a\u96c6&#8221; \u5230\u8f68\u9053$\\mathcal{O}_y$\u4e0a\u7684\u70b9\u9644\u8fd1.<\/p>\n<h3>4. \u76f8\u5173\u8bc1\u660e\u6ce8\u8bb0<\/h3>\n<h5>4.1 \u6ce8\u8bb01<\/h5>\n<p>\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u81ea\u7531\u7fa4\u4f5c\u7528\u4e0e\u7eaf\u4e0d\u8fde\u7eed\u8fd9\u4e24\u4e2a\u6982\u5ff5\u5f97\u5230\u4e00\u4e2a\u975e\u5e38\u6709\u7528\u7684\u547d\u9898.<br \/>\n<strong>\u547d\u98981<\/strong> \u8bbe$X$\u662f\u4e00\u4e2aHausdorff\u7a7a\u95f4, $q: G \\searrow X$\u662f\u4e00\u4e2a\u7eaf\u4e0d\u8fde\u7eed\u7684\u7fa4\u4f5c\u7528, \u5219\u8f68\u9053\u7a7a\u95f4$X \/ G$\u4e5f\u662fHausdorff\u7a7a\u95f4. \u5982\u679c$q$\u8fd8\u662f\u81ea\u7531\u7684, \u5219\u7c98\u5408\u6620\u5c04$$p: X \\to X \/ G, x \\mapsto \\mathcal{O}_x$$\u662f\u5c40\u90e8\u540c\u80da.<\/p>\n<p>\u5728\u4e66\u4e0aP227\u7684\u8bc1\u660e\u4e2d, \u6784\u9020\u4e86\u8f68\u9053\u7a7a\u95f4\u4e2d$\\mathcal{O}_x$\u4e0e$\\mathcal{O}_y$\u7684\u4e24\u4e2a\u4e0d\u76f8\u4ea4\u7684\u90bb\u57df, \u6b64\u5904\u7684\u6784\u9020\u5341\u5206\u5de7\u5999, \u82e5\u76f4\u63a5\u53d6$U_x, U_y$, \u662f\u65e0\u6cd5\u8bf4\u660e\u8f68\u9053\u7a7a\u95f4$X \/ G$\u7684Hausdorff\u6027\u8d28\u7684. \u56e0\u6b64, \u8bc1\u660e\u4e2d\u5229\u7528\u62d3\u6251\u7a7a\u95f4$X$\u7684Hausdorff\u6027\u8d28\u7684, \u5c06$U_x, U_y$\u8fdb\u4e00\u6b65&#8221;\u7f29\u5c0f&#8221;, \u5f97\u5230$W_x, W_y$, \u4f7f\u5f97$g( $$ W_x) \\cap W_y = \\emptyset$, $\\forall g \\in G$. \u5176\u4e2d, $x \\in W_x$, $y \\in W_y$, \u6545$W_x, W_y$\u603b\u4e0d\u4e3a\u7a7a, \u4ece\u800c\u8fd9\u6837\u7684\u6784\u9020\u662f\u6709\u610f\u4e49\u7684.<\/p>\n<h5>4.2 \u6ce8\u8bb02<\/h5>\n<p>\u5728\u4e66\u4e0aP229\u7684\u4f8b6\u8bba\u8ff0\u4e2d, \u5206\u522b\u6784\u9020\u4e86\u4e24\u4e2a\u4ece\u7d27\u81f4\u7a7a\u95f4$D^2$(\u5355\u4f4d\u5706\u76d8) \u5230Hausdorff\u7a7a\u95f4$A, B$\u7684\u8fde\u7eed\u6620\u5c04, \u56e0\u6b64\u90fd\u662f\u540c\u80da. \u800c\u4e14\u5b83\u4eec\u90fd\u628a\u5355\u4f4d\u5706\u76d8\u7684\u8fb9\u754c\u5706\u5468\u5206\u522b\u53d8\u5230Hausdorff\u7a7a\u95f4$A$\u7684\u8fb9\u754c\u4e0eHausdorff\u7a7a\u95f4$B$\u7684\u8fb9\u754c, \u800c\u5b83\u4eec\u7684\u8fb9\u754c\u6070\u4e3a$$A \\cap B = \\{ \\mathcal{O}_{(z_1, z_2)} \\ | \\ |z_1| = |z_2|\\}.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u5468\u5b85\u5bb6, \u60f3\u7740\u5c3d\u91cf\u628a\u7fa4\u4f5c\u7528\u8fd9\u4e00\u8282\u5185\u5bb9\u7ed9\u5b8c\u7ed3\u4e86~<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1810"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1810"}],"version-history":[{"count":21,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1810\/revisions"}],"predecessor-version":[{"id":3606,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1810\/revisions\/3606"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1810"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1810"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1810"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}