Skip to main content
SpringerLink
Log in

Matrix group integrals, surfaces, and mapping class groups I: U(n)

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Since the 1970’s, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group  \({\mathcal {U}}\left( n\right) \) of unitary matrices. More concretely, we study measures induced by free words on \({\mathcal {U}}\left( n\right) \). Let \({\mathbf {F}}_{r}\) be the free group on r generators. To sample a random element from  \({\mathcal {U}}\left( n\right) \) according to the measure induced by \(w\in {\mathbf {F}}_{r}\), one substitutes the r letters in w by r independent, Haar-random elements from \({\mathcal {U}}\left( n\right) \). The main theme of this paper is that every moment of this measure is determined by families of pairs \(\left( \Sigma ,f\right) \), where \(\Sigma \) is an orientable surface with boundary, and f is a map from \(\Sigma \) to the bouquet of r circles, which sends the boundary components of \(\Sigma \) to powers of w. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of \(\Sigma \). As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on \({\mathcal {U}}\left( n\right) \) bears information about the number of solutions to the equation \(\left[ u_{1},v_{1}\right] \cdots \left[ u_{g},v_{g}\right] =w\) in the free group, and deduce that one can “hear” the stable commutator length of a word through its unitary word measures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Notes

  1. Let us mention that the w-measure on \({\mathcal {U}}\left( n\right) \) is completely determined by moments of this type where the words are taken to be powers of w: \(\mathcal{T}r_{w^{\alpha _{1}},\ldots ,w^{\alpha _{\ell }}}\left( n\right) \) with \(\alpha _{1},\ldots ,\alpha _{\ell }\in {\mathbb {Z}}\). See, for example, [30, Sect. 2.2]. (We comment about the preprint [30] in Remark 1.15.)

  2. Every example in Table 1 satisfies that for every generator \(x_{i}\), the total number of occurrences in \(w_{1},\ldots ,w_{\ell }\) of \(x_{i}^{+1}\) is equal to the number of occurrences of \(x_{i}^{-1}\). The reason is the simple fact that otherwise \(\mathcal{T}r_{w_{1},\ldots ,w_{\ell }}\left( n\right) \) is constantly zero—see Claim 2.1 below.

  3. The “ordinary” Euler characteristic of a group is defined for a large class of groups of certain finiteness conditions—see [3, Chapter IX]. The simplest case is when a group \(\Gamma \) admits a finite CW-complex as Eilenberg-MacLane space of type \(K\left( \Gamma ,1\right) \), namely, a path-connected complex with fundamental group isomorphic to \(\Gamma \) and a contractible universal cover. In this case, the Euler characteristic of \(\Gamma \) coincides with the Euler characteristic of the \(K\left( \Gamma ,1\right) \)-space.

  4. As in Proposition 1.1, the equality (1.2) holds for every \(n\ge \max _{x\in B}L_{x}\), where \(L_{x}\) is the total number of appearance of \(x^{+1}\) in the words \(w_{1},\ldots ,w_{\ell }\). See also Sect. 2.

  5. A more general concept of the commutator length was introduced by Calegari (e.g. [5, Definition 2.71]), and applies to finite sets of words \(w_{1},\ldots ,w_{\ell }\). This number can be related, under certain restrictions, to \(\chi _{\max }\left( w_{1},\ldots ,w_{\ell }\right) \), in a similar fashion to the \(\ell =1\) case.

  6. The function \(\mathrm {M}{\ddot{\mathrm {o}}}\mathrm {b}\) is the Möbius function on a natural poset structure on \(S_{L}\)—see, for instance, [34, Lectures 10 and 23].

  7. We use \(\left| w\right| \) to denote the number of letters in w.

  8. Interestingly, very similar constraints on n appear in a formula giving the expected trace of w in r uniform \(n\times n\)permutation matrices as a rational expression in n—see [37, Sect. 5].

  9. Novaes [33] has recently obtained a combinatorial formula for the Weingarten function in terms of maps on surfaces; our approach here is different and incorporates that we are integrating over independent unitary matrices, which naturally leads to considerations about infinite groups.

  10. We mention this specifically because when \(\kappa _{x}=0\) this does not follow from the previous bullet points.

  11. We call \(\left[ g\right] \) the isotopy class of g, rather than the homotopy class, because if one thinks of g as a collection of disjoint colored arcs and curves embedded in \(\Sigma \), then \(\left[ g\right] \) is indeed the isotopy class of this collection relative to \(V_{o}\).

  12. Here, a rectangle is a disc bounded by two arcs and two pieces of \(\partial \Sigma \), and an annulus is bounded by two curves. Restriction 3 should resonate the constraint on the set of matchings \(\mathrm {\overline{MATCH}}^{\kappa }\left( w_{1},\ldots ,w_{\ell }\right) \) from Sect. 2.4. In particular, if \(g\left( \Sigma \right) \) does not contain the circle in \({{\bigvee }^{r}S^{1}}\) associated with x, then necessarily \(\kappa _{x}=0\).

  13. Recall Remark 1.14 that we assume \(w_{i}\ne 1\) throughout the proofs. If we do consider the case that some of the words are trivial, then \(\Sigma \) may contain components made of discs, and the bound in Lemma 3.12 needs to be updated.

  14. As we explained in the proof of Lemma 3.8, in the current scenario, a zone violating Restriction 1 is necessarily a zone bounded by curves all of which are of the same color. By removing the zone we mean removing all bounding curves to obtain a new transverse map, and this procedure does not change the homotopy type of the map relative to \(V_{o}\).

  15. If n is odd, the parallel claim is the analog of \(\left[ h''\right] \preceq _{\mathcal{L}}\left[ h\right] \).

  16. For n odd, \(r_{n}\left( \left[ h\right] \right) \preceq _{\mathcal{L}}\left[ h\right] \).

References

  1. Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 308(1505), 523–615 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amit, A., Vishne, U.: Characters and solutions to equations in finite groups. J. Algebra Appl. 10(4), 675–686 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York (1982)

    Book  MATH  Google Scholar 

  4. Calegari, D.: What is\(\ldots \) stable commutator length? Not. Am. Math. Soc. 55(9), 1100–1101 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Calegari, D.: scl, MSJ Memoirs, vol. 20. Mathematical Society of Japan, Tokyo (2009)

    MATH  Google Scholar 

  6. Calegari, D.: Stable commutator length is rational in free groups. J. Am. Math. Soc. 22(4), 941–961 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheeger, J., Gromov, M.: \(L_2\)-cohomology and group cohomology. Topology 25(2), 189–215 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 2003(17), 953–982 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Culler, M.: Using surfaces to solve equations in free groups. Topology 20(2), 133–145 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Diaconis, P., Shahshahani, M.: On the eigenvalues of random matrices. J. Appl. Probab. 31, 49–62 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Edmunds, C.C.: On the endomorphism problem for free groups. Commun. Algebra 3(1), 1–20 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  15. Fulton, W.: Young tableaux: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  16. Goldman, W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200–225 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goldman, W.M.: Ergodic theory on moduli spaces. Ann. Math. (2) 146(3), 475–507 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goldstein, R.Z., Turner, E.C.: Applications of topological graph theory to group theory. Math. Z. 165(1), 1–10 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harvey, W. J.: Boundary structure of the modular group, Riemann surfaces and related topics. In: Proceedings of the 1978 Stony Brook Conference, State University New York, Stony Brook, N.Y., 1978, Annals of Mathematics Studies, vol. 97, Princeton University Press, Princeton, pp. 245–251 (1981)

    Chapter  Google Scholar 

  20. Harer, J.L.: Stability of the homology of the mapping class groups of orientable surfaces. Ann. Math. (2) 121(2), 215–249 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Harer, J.L.: The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84(1), 157–176 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hui, C.Y., Larsen, M., Shalev, A.: The Waring problem for Lie groups and Chevalley groups. Isr. J. Math. 210(1), 81–100 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Harer, J.L., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85(3), 457–485 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ivanov, N.V.: Mapping Class Groups. Handbook of Geometric Topology, pp. 523–633. Elsevier, Amsterdam (2002)

    MATH  Google Scholar 

  25. Labourie, F.: Lectures on Representations of Surface Groups. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2013)

    Book  MATH  Google Scholar 

  26. Lück, W.: \(L^2\)-invariants: theory and applications to geometry and \(K\)-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer, Berlin (2002)

  27. Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences, vol. 141. Springer, Berlin (2004). (With an appendix by Don B. Zagier, Low-Dimensional Topology, II)

    Book  Google Scholar 

  28. McMullen, C.T.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. Second Ser. 151(1), 327–357 (2000). (eng)

    Article  MATH  Google Scholar 

  29. Mondello, G.: A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces. J. Eur. Math. Soc. (JEMS) 10(1), 231–241 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Magee, M., Puder, D.: Word measures on unitary groups, arXiv preprint 1509.07374v2 [math.GR] (2016)

  31. Magee, M., Puder, D.: Surface words are determined by word measures on groups (2019). arXiv:1902.04873

  32. Mingo, J.A., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209(1), 212–240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Novaes, M.: Expansion of polynomial lie group integrals in terms of certain maps on surfaces, and factorizations of permutations. J. Phys. A: Math. Theor. 50(7), 075201 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  35. Penner, R.C.: Perturbative series and the moduli space of Riemann surfaces. J. Differ. Geom. 27(1), 35–53 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Puder, D., Parzanchevski, O.: Measure preserving words are primitive. J. Am. Math. Soc. 28(1), 63–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Puder, D.: Primitive words, free factors and measure preservation. Isr. J. Math. 201(1), 25–73 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Quillen, D.: Higher Algebraic \(K\)-Theory. I. Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)

    MATH  Google Scholar 

  39. Rădulescu, F.: Combinatorial aspects of Connes’s embedding conjecture and asymptotic distribution of traces of products of unitaries. In: Proceedings of the Operator Algebra Conference, Bucharest, Theta Foundation (2006)

  40. Serre, J.P.: Cohomologie des groupes discrets. In: Prospects in Mathematics, Annals of Mathematics Studies, vol. 70, pp. 77–169. Princeton University Press, Princeton (1971)

    Chapter  Google Scholar 

  41. Shalev, A.: Some results and problems in the theory of word maps, Erdős Centennial (Bolyai Society Mathematical Studies). In: Lovász, L., Ruzsa, I., Sós, V.T., Palvolgyi, D. (eds.) Springer, pp. 611–650 (2013)

  42. tom Dieck, T.: Orbittypen und äquivariante Homologie. I. Arch. Math. (Basel) 23, 307–317 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  43. tom Dieck, T.: Transformation Groups. De Gruyter Studies in Mathematics, vol. 8. Walter de Gruyter & Co., Berlin (1987)

    Book  MATH  Google Scholar 

  44. ’t Hooft, G.: A planar diagram theory for strong interactions. Nuclear Phys. B 72(3), 461–473 (1974)

    Article  MathSciNet  Google Scholar 

  45. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups (1992)

  46. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  47. von Neumann, J.: Zur allgemeinen Theorie des Maßes. Fundam. Math. 13, 73–116 (1929). (German)

    Article  MATH  Google Scholar 

  48. Walker, J.W.: Canonical homeomorphisms of posets. Eur. J. Combin. 9(2), 97–107 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  49. Weingarten, D.: Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys. 19(5), 999–1001 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  50. Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141(1), 153–209 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  51. Xu, F.: A random matrix model from two dimensional Yang–Mills theory. Commun. Math. Phys. 190(2), 287–307 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Danny Calegari, Alexei Entin, Mark Feighn, Alex Gamburd, Yair Minsky, Mark Powell, Peter Sarnak, Zlil Sela, Karen Vogtmann, Alden Walker, Avi Wigderson, Qi You, and Ofer Zeitouni for valuable discussions about this work. Part of this research was carried out during research visits of the first named author (Magee) to the Institute for Advanced Study in Princeton, and we would like to thank the I.A.S. for making these visits possible.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Rd, Durham, DH1 3LE, UK

    Michael Magee

  2. School of Mathematical Sciences, Tel Aviv University, 6997801, Tel Aviv, Israel

    Doron Puder

Authors
  1. Michael Magee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Doron Puder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Doron Puder.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

M. Magee was partially supported by the N.S.F. via Grants DMS-1128155 and DMS-1701357.

D. Puder was partially supported by the Rothschild Fellowship, N.S.F. via Grant DMS-1128155, and I.S.F. via Grant 1071/16.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Magee, M., Puder, D. Matrix group integrals, surfaces, and mapping class groups I: U(n). Invent. math. 218, 341–411 (2019). https://doi.org/10.1007/s00222-019-00891-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-019-00891-4

Search

Navigation