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Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.
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This content will become publicly available on February 28, 2026
On Set Representation of Bounded Degree Hypergaphs
In their classical paper, Erdős, Goodman and Pósa studied the representation of a graph with vertex set $[n]$ by a family of subsets $$S_1,\dots, S_n$$ with the property that $$\{i,j\}$$ is an edge if and only if $$S_i\cap S_j\neq \emptyset$$. In this note, we consider a similar representation of bounded degree $$r$$-uniform hypergraphs and establish some bounds for a corresponding problem.
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- Award ID(s):
- 2300347
- PAR ID:
- 10584368
- Publisher / Repository:
- Electron. J. Combin.
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 32
- Issue:
- 1
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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