Contracting automorphisms and Lp-cohomology in degree one

Cornulier, Yves de; Tessera, Romain

doi:10.1007/s11512-010-0127-z

Cited by 21 publications

(40 citation statements)

References 14 publications

(26 reference statements)

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“…We first prove (i), which is a simple consequence of the structure of the boundary. We next prove (ii), using L p -cohomology computations from [CT11].…”

Section: Qi-invariance Of the -Invariantmentioning

confidence: 98%

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Metric Geometry of Locally Compact Groups

Cornulier

Harpe

2016

EMS Tracts in Mathematics

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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 favourably 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 (Birkhoff-Kakutani, Kakutani-Kodaira, Struble). Other results concern special classes of groups, for example those mapping onto Z (the Bieri-Strebel splitting theorem, generalized to 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, which generalize 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.

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“…We first prove (i), which is a simple consequence of the structure of the boundary. We next prove (ii), using L p -cohomology computations from [CT11].…”

Section: Qi-invariance Of the -Invariantmentioning

confidence: 98%

“…We use computations of L p -cohomology carried out in [CT11] (inspired by Pansu's computations for Lie groups). Recall that for any locally compact group G endowed with a left Haar measure, its first L p -cohomology group H p 1 (G) is defined as the 1-cohomology of its right regular representation.…”

Section: Qi-invariance Of the -Invariantmentioning

confidence: 99%

Metric Geometry of Locally Compact Groups

Cornulier

Harpe

2016

EMS Tracts in Mathematics

Self Cite

116

142

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“…For example, from the discrete case, the intuition has grown that hyperbolicity and amenability are somehow incompatible: it is known that non-elementary finitely generated hyperbolic groups contain a free non-abelian subgroup F 2 and are thus non-amenable. On the other hand, in [14], the authors prove the counter-intuitive fact that there do exist amenable non-elementary locally compact hyperbolic groups! There are plenty of other differences as well: for example, it is possible for locally compact hyperbolic groups to act transitively on their boundary, even if the boundary is infinite.…”

Section: Locally Compact Hyperbolic Groupsmentioning

confidence: 98%

Embeddings of locally compact hyperbolic groups into L-spaces

Cave

Dreesen

2016

Topology and its Applications

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In the last years, there has been a large amount of research on embeddability properties of finitely generated hyperbolic groups. In this paper, we elaborate on the more general class of locally compact hyperbolic groups. We compute the equivariant L p -compression in a number of locally compact examples, such as the groups SO(n, 1). Next, we show that although there are locally compact, non-discrete hyperbolic groups G with Kazhdan's property (T ), it is true that any locally compact hyperbolic group admits a proper affine isometric action on an L p -space for p larger than the Ahlfors regular conformal dimension of ∂G. This answers a question asked by Yves de Cornulier. Finally, we elaborate on the locally compact version of property (A) and show that, as in the discrete case, it is connected to the non-equivariant Hilbert space compression of a group.

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“…One should think of ℓ q,p cohomology as a (large scale) topological invariant. It has been useful in several contexts, mainly for the class of hyperbolic groups where the relevant value of q is q = p, see [2,3,5,6] for instance. It is interesting to study a class of spaces where values of q = p play a significant role.…”

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confidence: 99%

“…,(5) and |∇| µ+k = |∇| µ |∇| k on S for µ, k ∈ N as comes from[8, Theorem 3.15]. Density of S 0 and extension of K c…”

mentioning

confidence: 99%