Generalized cusps in real projective manifolds: classification

Ballas, Samuel A.; Cooper, Daryl; Leitner, Arielle

doi:10.1112/topo.12161

Cited by 13 publications

(30 citation statements)

References 32 publications

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“…We briefly review the most relevant aspects of the results in Ballas, Cooper, and Leitner's paper . First, we borrow the following definition Definition A generalized cusp is a properly convex projective manifold

C = Γ ∖ Ω

with

normalΓ

virtually abelian,

\partial C

compact and strictly convex, and

C

diffeomorphic to

\partial C \times {double-struckR}_{⩾ 0}

.…”

Section: Cusp Neighborhoods and Groupsmentioning

confidence: 99%

“…It is verifiable by a calculation that

H (ψ)

preserves

Ω (ψ)

, although it is not the full automorphism group of

Ω (ψ)

. Let the reader be aware that we have exchanged the roles of the first and

(t + 1)

st coordinates from , because it will suit our needs better in Section 4.2. We may now define a

ψ

‐cusp as a quotient of the model cusp neighborhood (see also Fig.…”

Section: Cusp Neighborhoods and Groupsmentioning

confidence: 99%

“…Intuitively, removing some compact portion of a cusp should not count as a different cusp. For a precise treatment of this equivalence, we refer the reader to . For our purposes, it will be sufficient to rely on the following theorem from the same.…”

Section: Cusp Neighborhoods and Groupsmentioning

confidence: 99%

“…Note that the sets

P_{c}

P ({double-struckR}^{n + 1})

given by

P_{c} = \{([] (c - \sum_{i = 2}^{t + 1} a_{i} \log (x_{i}) + \frac{1}{2} \sum_{i = t + 2}^{n} x_{i}^{2}), x_{2}, \dots, x_{n}, 1)\}

with

a_{i} = \frac{b_{i}^{2} false(e^{s_{i}} + 1 false)}{2 false(e^{s_{i}} - 1 false) false(s_{i} false)}

are preserved by all

γ_{i}^{'}

. Hence,

ρ_{scriptS} false(π_{1} P false)

is a subgroup of

H (ψ)

, and by [, Theorem 0.2],

d e v_{scriptS} false(P false)

is equivalent to a type‐

t

cusp corresponding to

ψ = (a_{1}, \dots, a_{t}, 0, \dots, 0)

(after reordering the first

t

coordinates so that

a_{i}

are non‐increasing).

□

…”

Section: Analysis Of Bending Cusp Neighborhoodsmentioning

confidence: 99%

“…The task of understanding convex projective cusps was initiated by Cooper, Long, and Tilmann . Recently, Ballas, Cooper, and Leitner gave a complete classification of convex projective cusps with compact cross‐section . The result of this classification is a stratified space of structures, the

(n + 1)

strata in dimension

n

referred to as types ,

t = 0, \dots, n

, of convex projective cusps.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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].

show abstract

C = Γ ∖ Ω

with

normalΓ

virtually abelian,

\partial C

compact and strictly convex, and

C

diffeomorphic to

\partial C \times {double-struckR}_{⩾ 0}

.…”

Section: Cusp Neighborhoods and Groupsmentioning

confidence: 99%

“…It is verifiable by a calculation that

H (ψ)

preserves

Ω (ψ)

, although it is not the full automorphism group of

Ω (ψ)

. Let the reader be aware that we have exchanged the roles of the first and

(t + 1)

st coordinates from , because it will suit our needs better in Section 4.2. We may now define a

ψ

‐cusp as a quotient of the model cusp neighborhood (see also Fig.…”

Section: Cusp Neighborhoods and Groupsmentioning

confidence: 99%

Section: Cusp Neighborhoods and Groupsmentioning

confidence: 99%

“…Note that the sets

P_{c}

P ({double-struckR}^{n + 1})

given by

P_{c} = \{([] (c - \sum_{i = 2}^{t + 1} a_{i} \log (x_{i}) + \frac{1}{2} \sum_{i = t + 2}^{n} x_{i}^{2}), x_{2}, \dots, x_{n}, 1)\}

with

a_{i} = \frac{b_{i}^{2} false(e^{s_{i}} + 1 false)}{2 false(e^{s_{i}} - 1 false) false(s_{i} false)}

are preserved by all

γ_{i}^{'}

. Hence,

ρ_{scriptS} false(π_{1} P false)

is a subgroup of

H (ψ)

, and by [, Theorem 0.2],

d e v_{scriptS} false(P false)

is equivalent to a type‐

t

cusp corresponding to

ψ = (a_{1}, \dots, a_{t}, 0, \dots, 0)

(after reordering the first

t

coordinates so that

a_{i}

are non‐increasing).

□

…”

Section: Analysis Of Bending Cusp Neighborhoodsmentioning

confidence: 99%

(n + 1)

strata in dimension

n

referred to as types ,

t = 0, \dots, n

, of convex projective cusps.…”

Section: Introductionmentioning

confidence: 99%

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

Bobb

2019

J. London Math. Soc.

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Divisible convex sets with properly embedded cones

Blayac,

Viaggi

2024

Publ.math.IHES

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In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.

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Gluing equations for real projective structures on 3-manifolds

Ballas

Casella

2021

Geom Dedicata

Self Cite

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Given an orientable ideally triangulated 3-manifold M, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on M. These equations represent a unifying framework for the classical Thurston gluing equations in hyperbolic geometry and their more recent counterparts in Anti-de Sitter and halfpipe geometry. Moreover, these equations can be used to detect properly convex structures on M. The paper also includes explicit examples where the equations are used to construct properly convex structures.

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Generalized cusps in real projective manifolds: classification

Cited by 13 publications

References 32 publications

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

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

Divisible convex sets with properly embedded cones

Gluing equations for real projective structures on 3-manifolds

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