## Conjugacy classes of symmetries in orthogonal groups

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- by Donald G. James PDF
- Proc. Amer. Math. Soc.
**125**(1997), 747-753 Request permission

## Abstract:

The number of conjugacy classes of symmetries in the integral orthogonal group of an indefinite $\Bbb Z$-lattice is determined. The results are applied to the extended Bianchi groups.## References

- A. G. Earnest and J. S. Hsia,
*Correction to: “Spinor norms of local integral rotations. II” [Pacific J. Math. 61 (1975), no. 1, 71–86; MR0404142 (53 #7946)]*, Pacific J. Math.**115**(1984), no. 2, 493–494. MR**765203**, DOI 10.2140/pjm.1984.115.493 - Dzh. Èlstrod, F. Gryuneval′d, and Dzh. Mennike,
*Discontinuous groups in three-dimensional hyperbolic space: analytic theory and arithmetic applications*, Uspekhi Mat. Nauk**38**(1983), no. 1(229), 119–147 (Russian). Translated from the English by A. N. Parshin. MR**693720** - Alexander J. Hahn and O. Timothy O’Meara,
*The classical groups and $K$-theory*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR**1007302**, DOI 10.1007/978-3-662-13152-7 - D. G. James,
*Representations by integral quadratic forms*, J. Number Theory**4**(1972), 321–329. MR**309871**, DOI 10.1016/0022-314X(72)90067-4 - Donald G. James,
*Integral sums of squares in algebraic number fields*, Amer. J. Math.**113**(1991), no. 1, 129–146. MR**1087804**, DOI 10.2307/2374824 - D.G. James and C. Maclachlan,
*Fuchsian subgroups of Bianchi groups,*Trans. Amer. Math. Soc.**348**(1996), 1989-2002. - D. G. James and S. M. Rosenzweig,
*Associated vectors in lattices over valuation rings*, Amer. J. Math.**90**(1968), 295–307. MR**220702**, DOI 10.2307/2373438 - O. T. O’Meara,
*Introduction to quadratic forms*, Die Grundlehren der mathematischen Wissenschaften, Band 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0152507**, DOI 10.1007/978-3-642-62031-7 - Allan Trojan,
*The integral extension of isometries of quadratic forms over local fields*, Canadian J. Math.**18**(1966), 920–942. MR**202701**, DOI 10.4153/CJM-1966-092-x - È. B. Vinberg,
*Reflective subgroups in Bianchi groups*, Selecta Math. Soviet.**9**(1990), no. 4, 309–314. Selected translations. MR**1078259** - L. Ya. Vulakh,
*Reflections in extended Bianchi groups*, Math. Proc. Cambridge Philos. Soc.**115**(1994), no. 1, 13–25. MR**1253279**, DOI 10.1017/S0305004100071887

## Additional Information

**Donald G. James**- Email: james@math.psu.edu
- Received by editor(s): October 18, 1995
- Additional Notes: The author was supported by NSA grant MDA904-94-H-2034 and NSF grant DMS-95-00533.
- Communicated by: Ronald M. Solomon
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 747-753 - MSC (1991): Primary 11E57, 11F06, 20G30
- DOI: https://doi.org/10.1090/S0002-9939-97-04005-7
- MathSciNet review: 1416090