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تسجيل مستخدم جديدهندسة المجموعات المحيطة محلياً والنمو المعادلاتي وشكل الكرات الكبيرة
Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls
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تأليف
Emmanuel Breuillard
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نحن نحصل على التطبيقات الحدودية لحجم الكرات الكبيرة في مجموعة متحدة محليا موحدة G مع نمو بولينومي. يتم ذلك عبر دراسة الهندسة الجغرافية ل G وتطوير قسم الأبحاث ل P. Pansus. على وجه الخصوص، نظرنا إلى أن أي مجموعة مثل G تكون متشابهة ضعيفة ببعض المجموعات الحقيقية المتصلة البسيطة S، وهي ظل G. كما أظهرنا أن الكرات الكبيرة في G لها شكل حدودي، أي بعد تطبيق تجزئة مناسبة، يتحولون إلى مجموعة محدودة تحديدية التي يمكن تفسيرها جغرافيا. ثم نناقش سرعة التقارب، ونعالج بعض الأمثلة ونعطي تطبيقا لنظرية الأرجودية. كما نجيب على سؤال بوراجو حول المتغيرات اليسارية ونحيط ببعض النتائج التي حصل عليها ستول حول عدم الحكمية لسلاسل النمو للمجموعات النيلبوتية.
We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansus thesis. In particular, we show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.
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