Properly convex bending of hyperbolic manifolds

Ludovic Marquis; Samuel A. Ballas

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Properly convex bending of hyperbolic manifolds

Samuel A. Ballas, Ludovic Marquis

Geometric Topology · Submitted 10 Sep 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume.

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,636 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geq 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume.	result 0.93	0.94	0.94	asserted 1.00
s1	We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups.	contribution 0.99	0.86	0.03	asserted 0.64
s2	We then use this result to show in each dimension $d\geq 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds.	result 0.97	0.95	0.15	asserted 1.00
s3	Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.	result 0.99	0.95	0.07	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v3

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full text
figures and tables
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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geq 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.

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5 arXiv papers cite this in 8 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 6 sentences without a confident label

Cites it as background 2 papers

For a review of bending techniques in the context of convex projective geometry, we refer the reader to [ref] and [ref] .

Convex projective manifolds with a cusp of any non-diagonalizable type · Bending representations · 0.96
There are some results on deforming cusped hyperbolic manifolds to change the cuspidal holonomy [ref] .

Codimension-$1$ Simplices in Divisible Convex Domains · Further question · 0.83

What it cites · 3 sentences without a confident label

Cites as background 2 papers

Both standard and bent cusps are examples of generalized cusps, introduced by Cooper–Long–Tillmann [ref] .

Deforming convex projective manifolds · Introduction · 0.93
As a result, much recent work has been focused on producing and understanding deformations of such manifold [ref] or the survey [ref] .

Deformations of convex real projective manifolds and orbifolds · Introduction · 0.98

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