ترغب بنشر مسار تعليمي؟ اضغط هنا

هندسة المجموعات المحيطة محلياً والنمو المعادلاتي وشكل الكرات الكبيرة

Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls

350   0   0.0 ( 0 )
 نشر من قبل Emmanuel Breuillard F.
 تاريخ النشر 2007
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study Infinite groups as geometric objects, as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favorably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. Word metrics for compactly generated groups play a major role. In the particular case of finitely generated groups, they were introduced by Dehn around 1910 in connection with the Word Problem. Some of the results exposed concern general locally compact groups, such as criteria for the existence of compatible metrics on locally compact groups. Other results concern special classes of groups, for example those mapping onto the group of integers (the Bieri-Strebel splitting theorem for locally compact groups). Prior to their applications to groups, the basic notions of coarse and large-scale geometry are developed in the general framework of metric spaces. Coarse geometry is that part of geometry concerning properties of metric spaces that can be formulated in terms of large distances only. In particular coarse connectedness, coarse simple connectedness, metric coarse equivalences, and quasi-isometries of metric spaces are given special attention. The final chapters are devoted to the more restricted class of compactly presented groups, generalizing finitely presented groups to the locally compact setting. They can indeed be characterized as those compactly generated locally compact groups that are coarsely simply connected.
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
The purpose of this survey is to describe how locally compact groups can be studied as geometric objects. We will emphasize the main ideas and skip or just sketch most proofs, often referring the reader to our much more detailed book arXiv:1403.3796
223 - Bruce Kleiner 2007
We give a new proof of Gromovs theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorf f space. This answers an open question by Anantharaman-Delaroche.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا