## Duke Mathematical Journal

### The complex volume of $\operatorname {SL}(n,\mathbb{C})$-representations of 3-manifolds

#### Abstract

For a compact $3$-manifold $M$ with arbitrary (possibly empty) boundary, we give a parameterization of the set of conjugacy classes of boundary-unipotent representations of $\pi_{1}(M)$ into $\operatorname {SL}(n,\mathbb{C})$. Our parameterization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmüller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann’s extended Bloch group $\widehat{\mathcal{B}}(\mathbb{C})$, and we use this to obtain an efficient formula for the Cheeger–Chern–Simons invariant, and, in particular, for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic $3$-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

#### Article information

Source
Duke Math. J., Volume 164, Number 11 (2015), 2099-2160.

Dates
Revised: 4 September 2014
First available in Project Euclid: 13 August 2015

https://projecteuclid.org/euclid.dmj/1439470580

Digital Object Identifier
doi:10.1215/00127094-3121185

Mathematical Reviews number (MathSciNet)
MR3385130

Zentralblatt MATH identifier
1335.57034

#### Citation

Garoufalidis, Stavros; Thurston, Dylan P.; Zickert, Christian K. The complex volume of $\operatorname {SL}(n,\mathbb{C})$ -representations of 3-manifolds. Duke Math. J. 164 (2015), no. 11, 2099--2160. doi:10.1215/00127094-3121185. https://projecteuclid.org/euclid.dmj/1439470580

#### References

• [1] A. C. Aitken. Determinants and Matrices, Oliver and Boyd, Edinburgh, 1939.
• [2] N. Bergeron, E. Falbel, and A. Guilloux, Tetrahedra of flags, volume and homology of $\operatorname{SL}(3)$, Geom. Topol. 18 1911–1971.
• [3] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system, I: The user language, J. Symbolic Comput. 24 (1997), 235–265.
• [4] D. Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006), 209–227.
• [5] J. Cheeger and J. Simons, “Differential characters and geometric invariants” in Geometry and Topology (College Park, Md., 1983/84), Lecture Notes in Math. 1167, Springer, Berlin, 50–80, 1985.
• [6] S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48–69.
• [7] M. Culler, N. M. Dunfield, and J. R. Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds, http://www.math.uic.edu/t3m/SnapPy.
• [8] T. Dimofte, M. Gabella, and A. B. Goncharov, K-decompositions and 3d gauge theories, preprint, arXiv:1301.0192v1 [hep-th].
• [9] J. Dupont, R. Hain, and S. Zucker, “Regulators and characteristic classes of flat bundles” in The Arithmetic and Geometry of Algebraic Cycles (Banff, Alb., 1998), CRM Proc. Lecture Notes 24, Amer. Math. Soc., Providence, 2000, 47–92.
• [10] J. L. Dupont and F. W. Kamber, On a generalization of Cheeger–Chern–Simons classes, Illinois J. Math. 34 (1990), 221–255.
• [11] E. Falbel, A spherical CR structure on the complement of the figure eight knot with discrete holonomy, J. Differential Geom. 79 (2008), 69–110.
• [12] E. Falbel and Q. Wang, A combinatorial invariant for spherical CR structures, Asian J. Math. 17 (2013), 391–422.
• [13] V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211.
• [14] S. Garoufalidis, M. Goerner, and C. K. Zickert, Gluing equations for $\operatorname{PGL}(n,\mathbb{C})$-representations of 3-manifolds, Algebr. Geom. Top. 15 (2015), 565–622.
• [15] S. Garoufalidis, M. Goerner, and C. K. Zickert, The Ptolemy field of 3-manifold representations, Algebr. Geom. Top. 15 (2015), 371–397.
• [16] S. Goette and C. Zickert, The extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 11 (2007), 1623–1635.
• [17] O. Goodman, Snap, version 1.11.3, http://www.ms.unimelb.edu.au/~snap.
• [18] P. Menal-Ferrer and J. Porti, Local coordinates for $\operatorname{SL}(n,\mathbf{C})$-character varieties of finite-volume hyperbolic 3-manifolds, Ann. Math. Blaise Pascal 19 (2012), 107–122.
• [19] W. D. Neumann, Extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 8 (2004), 413–474.
• [20] W. Neumann, “Realizing arithmetic invariants of hyperbolic $3$-manifolds” in Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory), Contemp. Math. 541, Amer. Math. Soc., Providence, 2011, 233–246.
• [21] W. D. Neumann and D. Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), 307–332.
• [22] C.-H. Sah, Homology of classical Lie groups made discrete, III, J. Pure Appl. Algebra 56 (1989), 269–312.
• [23] N. Steenrod, The Topology of Fibre Bundles, Princeton Math. Ser. 14, Princeton Univ. Press, Princeton, 1951.
• [24] A. A. Suslin, “Homology of $\mathrm{GL}_{n}$, characteristic classes and Milnor $K$-theory” in Algebraic $K$-theory, Number Theory, Geometry and Analysis (Bielefeld, 1982), Lecture Notes in Math. 1046, Springer, Berlin, 1984, 357–375.
• [25] A. A. Suslin, $K_{3}$ of a field, and the Bloch group (in Russian), Trudy Mat. Inst. Steklov. 183 (1990), 180–199, 229; English translation in Proc. Steklov Inst. Math. 1991, no. 4, 217–239.
• [26] W. P. Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University, Princeton, 1980, http://library.msri.org/books/gt3m.
• [27] C. K. Zickert, The volume and Chern-Simons invariant of a representation, Duke Math. J. 150 (2009), 489–532.
• [28] C. K. Zickert, The extended Bloch group and algebraic $K$-theory, J. Reine Angew. Math. 704 (2015), 21–54.
• [29] C. K. Zickert, The extended Bloch group and algebraic $K$-theory, preprint, arXiv:0910.4005 [math.GT].