Arithmetic groups of higher 𝑄-rank cannot act on 1-manifolds

Witte, Dave

doi:10.1090/s0002-9939-1994-1198459-5

Cited by 67 publications

(62 citation statements)

References 10 publications

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“…This result is well known. It follows from the work of Morris (Witte) in [18] (also see Ghys [9] and Burger and Monod [3]). The corollary above holds for a large family of lattices that contain two commuting sublattices that satisfy the hypothesis of Theorem 1.5.…”

Section: Kazhdan Groupsmentioning

confidence: 93%

C¹actions on the mapping class groups on the circle

Parwani

2008

Algebr. Geom. Topol.

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Section: Kazhdan Groupsmentioning

confidence: 93%

C¹actions on the mapping class groups on the circle

Parwani

2008

Algebr. Geom. Topol.

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“…T ] should be trivial, since any C 0 -action of such a on S 1 is trivial, by a result of D Witte [40].…”

Section: Comments and Open Questionsmentioning

confidence: 99%

The braided Ptolemy–Thompson group is finitely presented

Funar

Kapoudjian

2008

Geom. Topol.

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Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group T ] (and its companion T ) which is an extension of the Ptolemy-Thompson group T by the braid group B 1 on infinitely many strands. We prove that T ] is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups T ] and T are in the same relation with respect to each other as the braid groups B nC1 and B n , for infinitely many strands n. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable. 20F36, 57M07; 20F38, 20F05, 57N05 IntroductionThe first relationships between Thompson's groups and braid groups were brought to light in the article [25] by P Greenberg and V Sergiescu, which is devoted to the construction and the homological study of extensions of Thompson's groups F and T by the stable braid group B Thompson's groups are not tree automorphisms, but are induced by piecewise tree automorphisms [28]. Therefore, a natural question is to find a way of lifting those elements to automorphisms of an appropriate structure. The answer proposed by [28] and [22] is to lift them to mapping classes of homeomorphisms of particular surfaces. Indeed, both groups A T and B are mapping class groups of infinite surfaces which are thickenings of suitable regular trees; the surfaces are endowed with an extra structure that must be, not globally, but only asymptotically preserved by the mapping classeshence the notion of asymptotic mapping class group. This extra structure may consist of a decomposition of the surface into pairs of pants, hexagons, hexagons with punctures, and so on.The surface D ] considered for the construction of the asymptotic mapping class group T ] is the planar thickened binary tree, which is punctured along an infinite discrete subset of points. The extra structure consists of a decomposition into suitably punctured hexagons. The asymptotic mapping class group that one obtains this way is an extension T ] of T by the group of braids B 1 on infinitely many strands (corresponding to the punctures). Therefore, T ] is quite similar to, but simpler than A T .This new group T ] seems interesting and worthy of deeper study. Compared with B , the definition of T ] presents new features, for instance, the dependence on the extra structure is now clearly manifest. We can choose two sets of punctures leading to homeomorphic surfaces for which the associated groups are not isomorphic. We obtain that way another group T , which is a sort of twin brother of T ] . Although T ] and T share the same properties, they are different. Our main result is the following: T (see Penner [35]). The terminology used here for T is expected to stress on its link with the Penner-Ptolemy groupoid. The group T ] is essentially different from BV (and B ), being an extension by the whole group of braids, and not only the pure braids. Moreover, it is know...

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“…L'un des phénomènes principaux qui permettent de justifier cette affirmation est donné par un théorème obtenu par l'auteur dans [27], lequel généralise dans plusieurs directions des résultats contenus dans [4], [5], [11], [16], [34], [39] et [40] (valables toutefois sous des hypothèses de régularité plus faibles). Rappelons qu'un groupe topologique localement compact possède la propriété (T) de Kazhdan si toute représentation affine (isométrique) de sur un espace de Hilbert admet un vecteur invariant.…”

Section: Introductionunclassified

Arithmetic groups of higher 𝑄-rank cannot act on 1-manifolds

Cited by 67 publications

References 10 publications

C¹actions on the mapping class groups on the circle

C¹actions on the mapping class groups on the circle

The braided Ptolemy–Thompson group is finitely presented

Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle

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Arithmetic groups of higher 𝑄-rank cannot act on 1-manifolds

Cited by 67 publications

References 10 publications

C1actions on the mapping class groups on the circle

C1actions on the mapping class groups on the circle

The braided Ptolemy–Thompson group is finitely presented

Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle

Contact Info

Product

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About

C¹actions on the mapping class groups on the circle

C¹actions on the mapping class groups on the circle