## Algebraic & Geometric Topology

### Treewidth, crushing and hyperbolic volume

#### Abstract

The treewidth of a $3$–manifold triangulation plays an important role in algorithmic $3$–manifold theory, and so it is useful to find bounds on the treewidth in terms of other properties of the manifold. We prove that there exists a universal constant $c$ such that any closed hyperbolic $3$–manifold admits a triangulation of treewidth at most the product of $c$ and the volume. The converse is not true: we show there exists a sequence of hyperbolic $3$–manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 5 (2019), 2625-2652.

Dates
Revised: 21 January 2019
Accepted: 4 February 2019
First available in Project Euclid: 26 October 2019

https://projecteuclid.org/euclid.agt/1572055269

Digital Object Identifier
doi:10.2140/agt.2019.19.2625

Mathematical Reviews number (MathSciNet)
MR4023324

#### Citation

Maria, Clément; Purcell, Jessica S. Treewidth, crushing and hyperbolic volume. Algebr. Geom. Topol. 19 (2019), no. 5, 2625--2652. doi:10.2140/agt.2019.19.2625. https://projecteuclid.org/euclid.agt/1572055269

#### References

• I Agol, Small $3$–manifolds of large genus, Geom. Dedicata 102 (2003) 53–64
• S Arnborg, D G Corneil, A Proskurowski, Complexity of finding embeddings in a $k$–tree, SIAM J. Algebraic Discrete Methods 8 (1987) 277–284
• S Arnborg, A Proskurowski, D G Corneil, Forbidden minors characterization of partial $3$–trees, Discrete Math. 80 (1990) 1–19
• R Benedetti, C Petronio, Lectures on hyperbolic geometry, Springer (1992)
• D Bienstock, On embedding graphs in trees, J. Combin. Theory Ser. B 49 (1990) 103–136
• H L Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM J. Comput. 25 (1996) 1305–1317
• G Burde, H Zieschang, Knots, De Gruyter Studies in Mathematics 5, de Gruyter, Berlin (1985)
• B A Burton, Introducing Regina, the $3$–manifold topology software, Experiment. Math. 13 (2004) 267–272
• B A Burton, A new approach to crushing $3$–manifold triangulations, Discrete Comput. Geom. 52 (2014) 116–139
• B A Burton, R G Downey, Courcelle's theorem for triangulations, J. Combin. Theory Ser. A 146 (2017) 264–294
• B A Burton, C Maria, J Spreer, Algorithms and complexity for Turaev–Viro invariants, from “Automata, languages, and programming, I” (M M Halldórsson, K Iwama, N Kobayashi, B Speckmann, editors), Lecture Notes in Comput. Sci. 9134, Springer (2015) 281–293
• B A Burton, J Spreer, The complexity of detecting taut angle structures on triangulations, from “Proceedings of the Twenty-Fourth Annual ACM–SIAM Symposium on Discrete Algorithms” (S Khanna, editor), SIAM, Philadelphia (2012) 168–183
• B Courcelle, The monadic second-order logic of graphs, I: Recognizable sets of finite graphs, Inform. and Comput. 85 (1990) 12–75
• M Culler, N M Dunfield, M Goerner, J R Weeks, SnapPy: a computer program for studying the geometry and topology of $3$–manifolds, software (2016) Available at \setbox0\makeatletter\@url http://snappy.computop.org/ {\unhbox0
• H Edelsbrunner, Geometry and topology for mesh generation, Cambridge Monographs on Applied and Computational Mathematics 7, Cambridge Univ. Press (2001)
• W Fenchel, Elementary geometry in hyperbolic space, De Gruyter Studies in Mathematics 11, de Gruyter, Berlin (1989)
• D Futer, E Kalfagianni, J S Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008) 429–464
• F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239–1284 With an appendix by David Futer
• K Huszár, J Spreer, U Wagner, On the treewidth of triangulated $3$–manifolds, from “34th International Symposium on Computational Geometry” (B Speckmann, C D Tóth, editors), Leibniz Int. Proc. Inform. 99, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2018) art. id. 46
• W Jaco, J H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61–168
• W Jaco, J H Rubinstein, Layered-triangulations of $3$–manifolds, preprint (2006)
• W Jaco, E Sedgwick, Decision problems in the space of Dehn fillings, Topology 42 (2003) 845–906
• D A Každan, G A Margulis, A proof of Selberg's hypothesis, Mat. Sb. 75 (117) (1968) 163–168 In Russian; translated in Math. USSR-Sb. 4 (1968) 147–152
• B Kleiner, J Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008) 2587–2855
• T Kobayashi, Y Rieck, A linear bound on the tetrahedral number of manifolds of bounded volume (after Jørgensen and Thurston), from “Topology and geometry in dimension three” (W Li, L Bartolini, J Johnson, F Luo, R Myers, J H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc., Providence, RI (2011) 27–42
• C Maria, J Spreer, A polynomial time algorithm to compute quantum invariants of $3$–manifolds with bounded first Betti number, from “Proceedings of the Twenty-Eighth Annual ACM–SIAM Symposium on Discrete Algorithms” (P N Klein, editor), SIAM, Philadelphia (2017) 2721–2732
• G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton Univ. Press (1973)
• G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)
• G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)
• J S Purcell, Hyperbolic knot theory, book project (2018) Available at \setbox0\makeatletter\@url http://users.monash.edu/~jpurcell/hypknottheory.html {\unhbox0
• N Robertson, P D Seymour, Graph minors, II: Algorithmic aspects of tree-width, J. Algorithms 7 (1986) 309–322
• M Sakuma, J Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. 21 (1995) 393–439
• A Satyanarayana, L Tung, A characterization of partial $3$–trees, Networks 20 (1990) 299–322
• P D Seymour, R Thomas, Call routing and the ratcatcher, Combinatorica 14 (1994) 217–241
• D M Thilikos, M J Serna, H L Bodlaender, Constructive linear time algorithms for small cutwidth and carving-width, from “Algorithms and computation” (D T Lee, S-H Teng, editors), Lecture Notes in Comput. Sci. 1969, Springer (2000) 192–203
• W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0