اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدQuasi-actions and rough Cayley graphs for locally compact groups
191
0
0.0
(
0
)
تأليف
Pekka Salmi
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its rough Cayley graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its rough Cayley graph is amenable as a metric space.
قيم البحث
اقرأ أيضاً
A connected, locally finite graph $Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $Gamma$. Define the minimal degree of $G$ as the minimal
degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${rm Aut}(T_d)$ is $d$ for all $dgeq 2$.
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
We construct locally compact groups with no non-trivial Invariant Random Subgroups and no non-trivial Uniformly Recurrent Subgroups.
We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on tr
ees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.
The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the best hyperbolic action of a group as the largest element of t
his poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have largest hyperbolic actions, including for instance many 3-manifold groups and most mapping class groups. Our proofs use the quasi-trees of metric spaces of Bestvina--Bromberg--Fujiwara, among other tools. In addition, we give a complete characterization of the poset of hyperbolic actions of Anosov mapping torus groups, and we show that mapping class groups of closed surfaces of genus at least two have hyperbolic actions which are comparable only to the trivial action.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
