The Generalized Cuspidal Cohomology Problem

Bart, Anneke; Scannell, Kevin P.

doi:10.4153/cjm-2006-028-1

Cited by 3 publications

(6 citation statements)

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“…Note that both free products with amalgamation and HNN extensions are examples of graph of groups decompositions. It is possible to discuss bending along many submanifolds simultaneously by describing the fundamental group as a graph of groups, and checking a cocycle condition on the intersections of the submanifolds [6], [13]. However, we will be discussing bending deformations that are well-behaved with respect to one another, and hence we will not require a great deal of technology.…”

Section: Iterated Bendingmentioning

confidence: 99%

Convex projective manifolds with a cusp of any non‐diagonalizable type

Bobb

2019

J. London Math. Soc.

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Recent work of Ballas, Cooper, and Leitner identifies (n + 1) types of n-dimensional convex projective cusps, one of which is the standard hyperbolic cusp. Work of Ballas-Marquis and Ballas-Danciger-Lee [Ballas, Danciger and Lee, 'Convex projective structures on nonhyperbolic threemanifolds',Geom. Topol. 22 (2018) 1593-1646] give examples of these exotic (non-hyperbolic) type cusps in dimension 3. Here an extension of the techniques of Ballas-Marquis shows the existence of all cusp types in all dimensions, except diagonalizable (type n) [Ballas, Cooper and Leitner, 'Generalized cusps in real projective manifolds: classification', Preprint, 2017, arXiv:1710.03132; Ballas and Marquis, 'Properly convex bending of hyperbolic manifolds', Preprint, 2016, arXiv:1609.03046].

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Section: Iterated Bendingmentioning

confidence: 99%

Convex projective manifolds with a cusp of any non‐diagonalizable type

Bobb

2019

J. London Math. Soc.

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“…Very little else is known about this question in general, though there are a few other interesting examples in the literature; some rigid [13], and some admitting deformations [5,6,12,14] (occasionally only to first order).…”

Section: Introductionmentioning

confidence: 99%

“…Our interest in the stamping example began with [6], in which we found an infinitesimal deformation of the link complement 8 2 14 supported on a piecewise totally geodesic 2-complex that is not isotopic to an immersed totally geodesic surface. This complex contains a "singular geodesic" formed by the intersection of three 2-cells along their boundaries, arranged combinatorially like three pages meeting the binding of a book.…”

Section: Introductionmentioning

confidence: 99%

A note on stamping

Bart

Scannell

2006

Geom Dedicata

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The stamping deformation was defined by Apanasov as the first example of a deformation of the flat conformal structure on a hyperbolic 3-orbifold distinct from bending. We show that in fact the stamping cocycle is equal to the sum of three bending cocycles. We also obtain a more general result, showing that derivatives of geodesic lengths vanish at the base representation under deformations of the flat conformal structure of a finite-volume hyperbolic 3-orbifold.

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“…(b) Generalized bending associated with a collection of compact totally-geodesic submanifolds with boundary in M n , see [12] 8 , [124], [159], [13]. The idea of the generalized bending is that instead of considering fundamental groups of graphs of groups, one looks at the more general complexes of groups.…”

Section: Theorem 116 (D Johnson and J Millsonmentioning

confidence: 99%

“…For every Γ-conjugacy class [Π i ] in P, choose a representative Π i ⊂ Γ. Then Π will denote 13 the set of all these representatives Π i . By abusing the notation, we will refer to the set Π as the set of cusps of virtual rank ≥ 2 in Γ.…”

Section: A Bit Of Homological Algebramentioning

confidence: 99%

Kleinian Groups in Higher Dimensions

Kapovich

Geometry and Dynamics of Groups and Spaces

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Abstract. This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space H n for n ≥ 4. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of H 3 .To the memory of Sasha Reznikov IntroductionThe goal of this survey is to give an overview (mainly from the topological perspective) of the theory of Kleinian groups in higher dimensions. The survey grew out of a series of lectures I gave in the University of Maryland in the Fall of 1991. An early (much shorter) version of this paper appeared as the preprint [109]. In this survey I collect well-known facts as well as less-known and new results. Hopefully, this will make the survey interesting to both non-experts and experts. We also refer the reader to Tukia's short survey [217] of higher-dimensional Kleinian groups.There is a vast variety of Kleinian groups in higher dimensions: It appears that there is no hope for a comprehensive structure theory similar to the theory of discrete groups of isometries of H 3 . I do not know a good guiding principle for the taxonomy of higher-dimensional Kleinian groups. In this paper the higher-dimensional Kleinian groups are organized according to the topological complexity of their limit sets. In this setting one of the key questions that I will address is the interaction between the geometry and topology of the limit set and the algebraic and topological properties of the Kleinian group. This paper is organized as follows. In Section 2 we consider the most basic concepts of the theory of Kleinian groups, e.g. domain of discontinuity, limit set, geometric finiteness, etc. In Section 3 we discuss various ways to construct Kleinian groups and list the tools of the theory of Kleinian groups in higher dimensions. In Section 4 we review the homological algebra used in the paper. In Section 5 we state topological rigidity results of Farrell and Jones and the coarse compact core theorem for higher-dimensional Kleinian groups. In Section 6 we discuss various notions of equivalence between Kleinian groups: From the weakest (isomorphism) to the strongest (conjugacy). In Section 7 we consider groups with zero-dimensional limit sets; such groups are relatively well-understood. Convex-cocompact groups with 1-dimensional limit sets are discussed in Section 8. Although the topology of the limit sets of such groups is well-understood, their group-theoretic structure is a mystery. We know very little about Kleinian groups with higher-dimensional limit sets, thus we restrict the discussion to Kleinian groups whose limit sets are topological spheres (Section 9). We Date: February 2, 2008. 1 then discuss Ahlfors finiteness theorem and its failure in higher dimensions (Section 10). We then consider the representation varieties of Kleinian groups (Section 11). Lastly we discuss algebraic and topological constraints on Kleinian groups in higher dimensions (Section 12).Acknowledgments. During this work ...

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The Generalized Cuspidal Cohomology Problem

Cited by 3 publications

References 36 publications

Convex projective manifolds with a cusp of any non‐diagonalizable type

Convex projective manifolds with a cusp of any non‐diagonalizable type

A note on stamping

Kleinian Groups in Higher Dimensions

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