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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: 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 $ {\mathcal G}^n$ be the group of orientation-preserving isometries of one of the space forms, the $ (n - 1)$-sphere, Euclidean $ n$-space, hyperbolic $ n$-space. For $ n \geq 3$, we show that the commutator length of $ {\mathcal G}^n$ is 1; i.e., every element of $ {\mathcal G}^n$ is a commutator. We also show that every element of $ {\mathcal G}^n$ 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 $ \;n\!\mod 4$, the involution length of $ {\mathcal G}^n$ is either 2 or 3.

In the second part of the paper, we show that all pairs in $ SO^4$ are linked but that the generic pair in the orientation-preserving isometries of hyperbolic 4-space or in $ {\mathcal G}^n$, $ n \geq 5$, is not.


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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: https://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.