Space form isometries as commutators and products of involutions

Authors:
Ara Basmajian and Bernard Maskit

Journal:
Trans. Amer. Math. Soc. **364** (2012), 5015-5033

MSC (2010):
Primary 51M10; Secondary 30F40

Published electronically:
April 5, 2012

MathSciNet review:
2922617

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Abstract | References | Similar Articles | Additional Information

Abstract: In dimensions 2 and 3 it is well known that given two orientation-preserving hyperbolic isometries that generate a non-elementary group, one can find a triple of involutions so that each isometry can be expressed as a product of two of the three involutions; in this case, we say that the isometries are *linked*.

In this paper, we investigate the extent to which a pair of isometries in higher dimensions can be linked. This question separates naturally into two parts. In the first part, we determine the least number of involutions needed to express an isometry as a product, and give two applications of our results; the second part is devoted to the question of linking.

In general, the commutator (involution) length of a group element is the least number of elements needed to express that element as a product of commutators (involutions), and the commutator (involution) length of the group is the supremum over all commutator (involution) lengths. Let be the group of orientation-preserving isometries of one of the space forms, the -sphere, Euclidean -space, hyperbolic -space. For , we show that the commutator length of is 1; i.e., every element of is a commutator. We also show that every element of can be written as a product of two involutions, not necessarily orientation-preserving; and, depending on the particular space and on the congruence class of , the involution length of is either 2 or 3.

In the second part of the paper, we show that all pairs in are linked but that the generic pair in the orientation-preserving isometries of hyperbolic 4-space or in , , is not.

**1.**Alan F. Beardon,*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777****2.**W. M. Goldman,*Trace coordinates on Fricke spaces of some simple hyperbolic surfaces*, Chapter 15, pp. 611-684, Handbook of Teichmüller theory, vol. II (A Papadopoulos, ed.), IRMA Lectures in Mathematics and Physics, volume 13, European Mathematical Society (2008).**3.**Bernard Maskit,*Kleinian groups*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR**959135****4.**Samuel Pasiencier and Hsien-chung Wang,*Commutators in a semi-simple Lie group*, Proc. Amer. Math. Soc.**13**(1962), 907–913. MR**0169947**, 10.1090/S0002-9939-1962-0169947-4**5.**K. M. Puri, Factoring of isometries of hyperbolic 4-space and a discreteness condition, Thesis, Rutgers University 2009.**6.**Rimhak Ree,*Commutators in semi-simple algebraic groups*, Proc. Amer. Math. Soc.**15**(1964), 457–460. MR**0161944**, 10.1090/S0002-9939-1964-0161944-X**7.**R. C. Thompson,*Commutators in the special and general linear groups*, Trans. Amer. Math. Soc.**101**(1961), 16–33. MR**0130917**, 10.1090/S0002-9947-1961-0130917-7**8.**María J. Wonenburger,*Transformations which are products of two involutions*, J. Math. Mech.**16**(1966), 327–338. MR**0206025**

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Additional Information

**Ara Basmajian**

Affiliation:
Department of Mathematics, Graduate Center and Hunter College, CUNY, New York, New York 10065

Email:
abasmajian@gc.cuny.edu

**Bernard Maskit**

Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, New York 11794

Email:
bernie@math.sunysb.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05639-X

Received by editor(s):
January 18, 2011

Received by editor(s) in revised form:
May 30, 2011

Published electronically:
April 5, 2012

Additional Notes:
The first author was supported in part by a PSC-CUNY Grant

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.