Home On conjugation orbits of semisimple pairs in rank one
Article
Licensed
Unlicensed Requires Authentication

On conjugation orbits of semisimple pairs in rank one

  • Krishnendu Gongopadhyay ORCID logo EMAIL logo and Sagar B. Kalane
Published/Copyright: June 13, 2019

Abstract

We consider the Lie groups SU(n,1) and Sp(n,1) that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in Sp(n,1) and SU(n,1) up to conjugacy. This gives local parametrization of the representations ρ in Hom(F2,G)/G such that both ρ(x) and ρ(y) are semisimple elements in G, where F2=x,y, G=Sp(n,1) or SU(n,1). We use the PSp(n,1)-configuration space M(n,i,m-i) of ordered m-tuples of distinct points in 𝐇n¯, where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on M(n,i,m-i).


Communicated by Anna Wienhard


Funding statement: The first-named author acknowledges partial support from SERB-DST MATRICS project number MTR/2017/000355.

Acknowledgements

We thank the referee for many comments and suggestions. Thanks are also due to John Parker for useful discussions on a preliminary draft of this paper during a visit to the ICTS Bangalore for participating in the programs ICTS/ggd2017/11 and ICTS/SGGS2017/11. We thank the ICTS for the hospitality and support during the visit. The second-named author thanks a UGC research fellowship for supporting him throughout this project.

References

[1] B. N. Apanasov and I. Kim, Cartan’s angular invariant and deformations in symmetric spaces of rank 1, Mat. Sb. 198 (2007), no. 2, 147–169. 10.1070/SM2007v198n02ABEH003832Search in Google Scholar

[2] U. Brehm, The shape invariant of triangles and trigonometry in two-point homogeneous spaces, Geom. Dedicata 33 (1990), no. 1, 59–76. 10.1007/BF00147601Search in Google Scholar

[3] U. Brehm and B. Et-Taoui, Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces, Manuscripta Math. 96 (1998), no. 1, 81–95. 10.1007/s002290050055Search in Google Scholar

[4] U. Brehm and B. Et-Taoui, Congruence criteria for finite subsets of quaternionic elliptic and quaternionic hyperbolic spaces, Geom. Dedicata 84 (2001), no. 1–3, 261–269. 10.1023/A:1010306423797Search in Google Scholar

[5] W. Cao, Congruence classes of points in quaternionic hyperbolic space, Geom. Dedicata 180 (2016), 203–228. 10.1007/s10711-015-0099-zSearch in Google Scholar

[6] W. Cao, The moduli space of points in quaternionic projective space, preprint (2017), https://arxiv.org/abs/1705.06458. 10.32917/h2020068Search in Google Scholar

[7] S. S. Chen and L. Greenberg, Hyperbolic spaces Contrib. Anal. (1974), 49–87. 10.1016/B978-0-12-044850-0.50013-7Search in Google Scholar

[8] H. Cunha, F. Dutenhefner, N. Gusevskii and R. S. Thebaldi, The moduli space of complex geodesics in the complex hyperbolic plane, J. Geom. Anal. 22 (2012), no. 2, 295–319. 10.1007/s12220-010-9189-1Search in Google Scholar

[9] H. Cunha and N. Gusevskii, On the moduli space of quadruples of points in the boundary of complex hyperbolic space, Transform. Groups 15 (2010), no. 2, 261–283. 10.1007/s00031-010-9086-5Search in Google Scholar

[10] H. Cunha and N. Gusevskii, The moduli space of points in the boundary of complex hyperbolic space, J. Geom. Anal. 22 (2012), no. 1, 1–11. 10.1007/s12220-010-9188-2Search in Google Scholar

[11] D. Z. Djoković, Poincaré series of some pure and mixed trace algebras of two generic matrices, J. Algebra 309 (2007), no. 2, 654–671. 10.1016/j.jalgebra.2006.09.018Search in Google Scholar

[12] E. Falbel, Geometric structures associated to triangulations as fixed point sets of involutions, Topology Appl. 154 (2007), no. 6, 1041–1052. 10.1016/j.topol.2006.10.006Search in Google Scholar

[13] E. Falbel and I. D. Platis, The PU(2,1) configuration space of four points in S3 and the cross-ratio variety, Math. Ann. 340 (2008), no. 4, 935–962. 10.1007/s00208-007-0177-0Search in Google Scholar

[14] W. M. Goldman, Complex Hyperbolic Geometry, Oxford Math. Monogr., Clarendon Press, New York, 1999. 10.1093/oso/9780198537939.001.0001Search in Google Scholar

[15] W. M. Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, Handbook of Teichmüller Theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, European Mathematical Society, Zürich (2009), 611–684. 10.4171/055-1/16Search in Google Scholar

[16] K. Gongopadhyay, The z-classes of quaternionic hyperbolic isometries, J. Group Theory 16 (2013), no. 6, 941–964. 10.1515/jgt-2013-0013Search in Google Scholar

[17] K. Gongopadhyay and S. B. Kalane, Quaternionic hyperbolic Fenchel–Nielsen coordinates, Geom. Dedicata 199 (2019), 247–271. 10.1007/s10711-018-0347-0Search in Google Scholar

[18] K. Gongopadhyay and S. Lawton, Invariants of pairs in SL(4,) and SU(3,1), Proc. Amer. Math. Soc. 145 (2017), no. 11, 4703–4715. 10.1090/proc/13638Search in Google Scholar

[19] K. Gongopadhyay and S. Parsad, Conjugation orbits of loxodromic pairs in SU(n,1), Bull. Sci. Math. 148 (2018), 14–32. 10.1016/j.bulsci.2018.06.004Search in Google Scholar

[20] K. Gongopadhyay and S. Parsad, On Fenchel–Nielsen coordinates of surface group representations into SU(3,1), Math. Proc. Cambridge Philos. Soc. 165 (2018), no. 1, 1–23. 10.1017/S0305004117000159Search in Google Scholar

[21] G. Gou and Y. Jiang, The moduli space of points in the boundary of quaternionic hyperbolic space, preprint (2017), https://arxiv.org/abs/1712.09217. Search in Google Scholar

[22] N. Gusevskii, The invariants of finite configurations in complex hyperbolic geometry, ICTP Lecture Notes in Advanced School: Geometry of Discrete Actions, 2010. Search in Google Scholar

[23] J. Hakim and H. Sandler, The moduli space of n+1 points in complex hyperbolic n-space, Geom. Dedicata 97 (2003), 3–15. 10.1023/A:1023616007798Search in Google Scholar

[24] R. Höfer, m-point invariants of real geometries, Beitr. Algebra Geom. 40 (1999), no. 1, 261–266. Search in Google Scholar

[25] I. Kim and J. R. Parker, Geometry of quaternionic hyperbolic manifolds, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 291–320. 10.1017/S030500410300687XSearch in Google Scholar

[26] S. Lawton, Generators, relations and symmetries in pairs of 3×3 unimodular matrices, J. Algebra 313 (2007), no. 2, 782–801. 10.1016/j.jalgebra.2007.01.003Search in Google Scholar

[27] S. Lawton, Minimal affine coordinates for SL(3,) character varieties of free groups, J. Algebra 320 (2008), no. 10, 3773–3810. 10.1016/j.jalgebra.2008.06.031Search in Google Scholar

[28] J. R. Parker, Traces in complex hyperbolic geometry, Geometry, Topology and Dynamics of character Varieties, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 23, World Scientific, Hackensack (2012), 191–245. 10.1142/9789814401364_0006Search in Google Scholar

[29] J. R. Parker and I. D. Platis, Complex hyperbolic Fenchel–Nielsen coordinates, Topology 47 (2008), no. 2, 101–135. 10.1016/j.top.2007.08.001Search in Google Scholar

[30] I. D. Platis, Cross-ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1, Geom. Dedicata 169 (2014), 187–208. 10.1007/s10711-013-9850-5Search in Google Scholar

[31] C. Procesi, The invariant theory of n×n matrices, Adv. Math. 19 (1976), no. 3, 306–381. 10.1016/0001-8708(76)90027-XSearch in Google Scholar

[32] L. Rodman, Topics in Quaternion Linear Algebra, Princeton Ser. Appl. Math., Princeton University, Princeton, 2014. 10.23943/princeton/9780691161853.001.0001Search in Google Scholar

[33] P. Will, Traces, cross-ratios and 2-generator subgroups of SU(2,1), Canad. J. Math. 61 (2009), no. 6, 1407–1436. 10.4153/CJM-2009-067-6Search in Google Scholar

Received: 2018-09-17
Revised: 2019-03-04
Published Online: 2019-06-13
Published in Print: 2019-09-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2018-0221/pdf
Scroll to top button