共查询到20条相似文献,搜索用时 890 毫秒
1.
We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of a finitely generated group with a continuum of non-π1-equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true. 相似文献
2.
Kei Funano 《Geometriae Dedicata》2010,149(1):103-119
In this paper we study actions of compact groups and of Lévy groups on a large class of metric spaces, such as
\mathbbR{\mathbb{R}} -trees, doubling spaces, metric graphs, and Hadamard manifolds, from the viewpoint of the theory of concentration of maps. 相似文献
3.
Brian H. Bowditch 《Geometriae Dedicata》2014,170(1):157-176
We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of $\mathbb{R }$ -trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete $\mathbb{R }$ -trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Dru?u and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of $\mathbb{R }$ -trees. 相似文献
4.
Matthias Hamann 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2017,87(2):245-263
We look at group actions on graphs and other metric spaces, e. g., at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results about the density of the hyperbolic limit set of the group in the whole limit set of the group. 相似文献
5.
Christoph Wockel 《Journal of Functional Analysis》2007,251(1):254-288
In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac-Moody algebras and groups. 相似文献
6.
We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of
a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral
subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry
invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce
a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely
generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin
groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic,
in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are
the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions
of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces
of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity
result in low complexity. 相似文献
7.
Non-Nesting Actions On Real Trees 总被引:1,自引:0,他引:1
The theory of isometric group actions on R-trees is extendedto actions by homeomorphisms with the following non-nestingproperty: no group element maps an arc properly into itself.A finitely presented group acting freely by homeomorphisms onan R-tree is free abelian or splits over a (possibly trivial)cyclic group. 1991 Mathematics Subject Classification 20E08,20F32, 57M60. 相似文献
8.
Thomas Delzant 《Geometric And Functional Analysis》2016,26(1):160-187
Let X be a compact Kähler manifold, and g a fixed genus. Due to the work of Parshin and Arakelov, it is known that there are only a finite number of non isotrivial holomorphic families of Riemann surfaces of genus \({g \geqslant 2}\) over X. We prove that this number only depends on the fundamental group of X. Our approach uses geometric group theory (limit groups, \({\mathbb{R}}\)-trees, the asymptotic geometry of the mapping class group), and Gromov-Shoen theory. We prove that in many important cases limit groups (in the sense of Sela) associated to infinite sequences of actions of a Kähler group on a Gromov-hyperbolic space are surface groups and we apply this result to monodromy groups acting on complexes of curves. 相似文献
9.
In this paper we use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard spaces, we formulate criterions for the action to have a global fixed point.Dedicated to Professor Takushiro Ochiai on his 60th birthday. 相似文献
10.
J. Higes 《Topology and its Applications》2010,157(17):2635-2645
In this paper we study two problems concerning Assouad-Nagata dimension:
- (1)
- Is there a metric space of positive asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]).
- (2)
- Suppose G is a locally finite group with a proper left invariant metric dG. If dimAN(G,dG)>0, is dimAN(G,dG) infinite? (Brodskiy et al., preprint [6, Problem 5.3]).
11.
Ronald G. Douglas 《Journal of Functional Analysis》2011,261(12):3723-3734
We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf G-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf G-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on ?∞-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras. 相似文献
12.
Hiroaki Nakamura Naotake Takao 《Transactions of the American Mathematical Society》1998,350(3):1079-1102
In this paper, Grothendieck's anabelian conjecture on the pro- fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This is done by estimating effectively the Galois equivariant automorphism group of the pro- braid group on the curve. The process of the proof involves the complete determination of the groups of graded automorphisms of the graded Lie algebras associated to the weight filtration of the braid groups on Riemann surfaces.
13.
Femke Douma 《Discrete Mathematics》2011,(4):276
Huber (1956) [8] considered the following problem on the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x∈H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree. 相似文献
14.
François Gautero 《Comptes Rendus Mathematique》2003,336(11):883-888
After introducing the notion of group automorphism hyperbolic relative to a family of subgroups, we establish an analog of the Bestvina–Feighn's Combination Theorem for mapping-tori groups of relatively hyperbolic automorphisms α of hyperbolic groups G. Both Farb's and Gromov's relative hyperbolicity are considered. To cite this article: F. Gautero, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
15.
B. H. Bowditch 《Transactions of the American Mathematical Society》1999,351(9):3673-3686
We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.
16.
V. Guirardel 《Commentarii Mathematici Helvetici》1998,73(1):89-121
This article shows how to approximate a stable action of a finitely presented group on an -tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an -tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on -trees and on the dynamical study of exotic components by D. Gaboriau.
Received: 22 October, 1997 相似文献
17.
Stephen M Paneitz 《Journal of Functional Analysis》1981,43(3):313-359
The convex cones in a simple Lie algebra invariant under the adjoint group G of are studied. Using a earlier abstract classification of such cones, we find explicit algebraic presentations of such cones in all the classical hermitian symmetric Lie algebras. (Nontrivial such cones exist only in these cases.) The G-orbits in such cones are listed. The notion of a temporal action of a Lie group with an invariant causal orientation upon a causally oriented manifold is defined. The canonical actions of such classical groups G as above on the S?hilov boundaries of the associated (tube-type) hermitian symmetric spaces are shown to be temporal actions. Corollaries are (1) the existence of nontrivial (Lie) semigroups S in the infinite-sheeted coverings of G, which are invariant under conjugation by and satisfy S ∩ S?1 = {e}, and (2) the global causality (i.e. no “closed time-like curves”) of such covering groups . 相似文献
18.
In this paper, we prove that every element of the linear group GL14(R) normalizing the Chevalley group of type G 2 over a commutative local ring R without 1/2 belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type G 2 over a local ring without 1/2 is a composition of a ring and an inner automorphisms. 相似文献
19.
Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries 总被引:1,自引:0,他引:1
We study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SL
n
(), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SL
n
() for n 3 cannot act on a visibility space X without fixing a point in
. Corollaries concern Floyd's group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included. 相似文献
20.
We present the construction of a kind of convex core for the product of two actions of a group on R-trees. This geometric construction allows one to generalize and unify the intersection number of two curves or of two measured foliations on a surface, Scott's intersection number of two splittings, and the appearance of surfaces in Fujiwara-Papasoglu's construction of the JSJ splitting. In particular, this construction allows a topological interpretation of the intersection number analogous to the definition for curves in surfaces. As an application of this construction, we prove that an irreducible automorphism of the free group whose stable and unstable trees are geometric, is actually induced by a pseudo-Anosov homeomorphism of a surface. 相似文献