## Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold

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- by D. Burago, S. Ferleger, B. Kleiner and A. Kononenko PDF
- Proc. Amer. Math. Soc.
**129**(2001), 1493-1498 Request permission

## Abstract:

Let $S$ be a smooth 3-dimensional nonpositively curved Riemannian manifold with corners, whose boundary consists of a finite number of geodesically convex nonpositively curved faces (for example, a Euclidean or hyperbolic polyhedron). We show that it is always possible to glue together finitely many copies of $S$ so as to get a nonpositively curved pseudomanifold without boundary.## References

- Stephanie B. Alexander and Richard L. Bishop,
*The Hadamard-Cartan theorem in locally convex metric spaces*, Enseign. Math. (2)**36**(1990), no. 3-4, 309–320. MR**1096422** - Werner Ballmann,
*Lectures on spaces of nonpositive curvature*, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR**1377265**, DOI 10.1007/978-3-0348-9240-7 - D. Burago, S. Ferleger, and A. Kononenko,
*Uniform estimates on the number of collisions in semi-dispersing billiards*, Ann. of Math. (2)**147**(1998), no. 3, 695–708. MR**1637663**, DOI 10.2307/120962 - D. Burago, S. Ferleger, and A. Kononenko,
*Topological entropy of semi-dispersing billiards*, Ergodic Theory Dynam. Systems**18**(1998), no. 4, 791–805. MR**1645377**, DOI 10.1017/S0143385798108246 - D. Burago, S. Ferleger, and A. Kononenko,
*Unfoldings and global bounds on the number of collisions for generalized semi-dispersing billiards*, Asian J. Math.**2**(1998), no. 1, 141–152. MR**1656555**, DOI 10.4310/AJM.1998.v2.n1.a4 - É. Ghys and P. de la Harpe (eds.),
*Sur les groupes hyperboliques d’après Mikhael Gromov*, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR**1086648**, DOI 10.1007/978-1-4684-9167-8 - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - John Hempel,
*Residual finiteness for $3$-manifolds*, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 379–396. MR**895623** - M. Kapovich.
*Hyperbolic manifolds and Discrete Groups. Notes on Thurston’s Hyperbolization.*University of Utah Lecture Notes, 1995. To appear in Birkhäuser, Progress in Mathematics. - Leonard Eugene Dickson,
*New First Course in the Theory of Equations*, John Wiley & Sons, Inc., New York, 1939. MR**0000002** - C. McMullen,
*Iteration on Teichmüller space*, Invent. Math.**99**(1990), no. 2, 425–454. MR**1031909**, DOI 10.1007/BF01234427 - Jean-Pierre Otal,
*Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3*, Astérisque**235**(1996), x+159 (French, with French summary). MR**1402300** - Jean-Pierre Otal,
*Thurston’s hyperbolization of Haken manifolds*, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 77–194. MR**1677888** *Geometry. IV*, Encyclopaedia of Mathematical Sciences, vol. 70, Springer-Verlag, Berlin, 1993. Nonregular Riemannian geometry; A translation of*Geometry, 4 (Russian)*, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [ MR1099201 (91k:53003)]; Translation by E. Primrose. MR**1263963**, DOI 10.1007/978-3-662-02897-1- Edwin H. Spanier,
*Algebraic topology*, Springer-Verlag, New York, [1995?]. Corrected reprint of the 1966 original. MR**1325242** - John Stillwell,
*Classical topology and combinatorial group theory*, 2nd ed., Graduate Texts in Mathematics, vol. 72, Springer-Verlag, New York, 1993. MR**1211642**, DOI 10.1007/978-1-4612-4372-4 - David A. Stone,
*Geodesics in piecewise linear manifolds*, Trans. Amer. Math. Soc.**215**(1976), 1–44. MR**402648**, DOI 10.1090/S0002-9947-1976-0402648-8 - William P. Thurston,
*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357–381. MR**648524**, DOI 10.1090/S0273-0979-1982-15003-0

## Additional Information

**D. Burago**- Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
- Email: burago@math.psu.edu
**S. Ferleger**- Affiliation: Renaissance Technology Corporation, 600 Rt. 25-A, East Setanket, New York 11787
**B. Kleiner**- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090
- Email: bkleiner@math.utah.edu
**A. Kononenko**- Affiliation: Renaissance Technology Corporation, 600 Rt. 25-A, East Setanket, New York 11787
- Email: kononena@yahoo.com
- Received by editor(s): August 31, 1998
- Published electronically: January 8, 2001
- Additional Notes: The first author was partially supported by a Sloan Foundation Fellowship and NSF grant DMS-9803129. The second author was partially supported by a Sloan Dissertation Fellowship. The third author was supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS-96-26911. The fourth author was supported by NSF grant DMS-9803092
- Communicated by: Christopher Croke
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**129**(2001), 1493-1498 - MSC (2000): Primary 51K10, 53C20; Secondary 52B10
- DOI: https://doi.org/10.1090/S0002-9939-01-05554-X
- MathSciNet review: 1707510