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Pacific Journal of Mathematics Vol. 249 (2011), No. 1 Pacific Journal of Mathematics Volume 249 No. 1 January 2011 PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982) and F. Wolf (1904–1989) EDITORS V. S. Varadarajan (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 pacifi[email protected] Vyjayanthi Chari Darren Long Sorin Popa Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Riverside, CA 92521-0135 Santa Barbara, CA 93106-3080 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] Robert Finn Jiang-Hua Lu Jie Qing Department of Mathematics Department of Mathematics Department of Mathematics Stanford University The University of Hong Kong University of California Stanford, CA 94305-2125 Pokfulam Rd., Hong Kong Santa Cruz, CA 95064 fi[email protected] [email protected] [email protected] Kefeng Liu Alexander Merkurjev Jonathan Rogawski Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] PRODUCTION pacifi[email protected] Silvio Levy, Scientific Editor Matthew Cargo, Senior Production Editor SUPPORTING INSTITUTIONS ACADEMIA SINICA, TAIPEI STANFORD UNIVERSITY UNIV. OF CALIF., SANTA CRUZ CALIFORNIA INST. OF TECHNOLOGY UNIV. OF BRITISH COLUMBIA UNIV. OF MONTANA INST. DE MATEMÁTICA PURA E APLICADA UNIV. OF CALIFORNIA, BERKELEY UNIV. OF OREGON KEIO UNIVERSITY UNIV. OF CALIFORNIA, DAVIS UNIV. OF SOUTHERN CALIFORNIA MATH. SCIENCES RESEARCH INSTITUTE UNIV. OF CALIFORNIA, LOS ANGELES UNIV. OF UTAH NEW MEXICO STATE UNIV. UNIV. OF CALIFORNIA, RIVERSIDE UNIV. OF WASHINGTON OREGON STATE UNIV. UNIV. OF CALIFORNIA, SAN DIEGO WASHINGTON STATE UNIVERSITY UNIV. OF CALIF., SANTA BARBARA These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. See inside back cover or www.pjmath.org for submission instructions. The subscription price for 2011 is US $420/year for the electronic version, and $485/year for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. Prior back issues are obtainable from Periodicals Service Company, 11 Main Street, Germantown, NY 12526-5635. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and the Science Citation Index. The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 969 Evans Hall, Berkeley, CA 94720-3840, is published monthly except July and August. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PJM peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS at the University of California, Berkeley 94720-3840 A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2011 by Pacific Journal of Mathematics PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011 METABELIAN SL.n; C/ REPRESENTATIONS OF KNOT GROUPS, II: FIXED POINTS HANS U. BODEN AND STEFAN FRIEDL Given a knot K in an integral homology sphere 6 with exterior NK , there is a natural action of the cyclic group Z=n on the space of SL.n; C/ represen- tations of the knot group π1.NK /, which induces an action on the SL.n; C/ character variety. We identify the fixed points of this action in terms of characters of metabelian representations, and we apply this in order to show α that the twisted Alexander polynomial 1K;1.t/ associated to an irreducible metabelian SL.n; C/ representation α is actually a polynomial in tn. 1. Introduction The study of metabelian representations and metabelian quotients of knot groups goes back to the pioneering work of Neuwirth [1965], de Rham [1967], Burde [1967], and Fox [1970]; see also [Burde and Zieschang 2003, Section 14]. The theory was further developed by many authors, including Hartley [1979; 1983], Livingston [1995], Letsche [2000], Lin [2001], Nagasato [2007], and Jebali [2008]. In [Boden and Friedl 2008], we proved a classification theorem for irreducible metabelian representations and in this paper we continue our study of metabelian representations of knot groups. Throughout this paper, when we say that K is a knot, we will always understand that K is an oriented, simple closed curve in an integral homology 3-sphere 6. We 3 write NK D 6 n τ.K /, where τ.K / denotes an open tubular neighborhood of K . Given a topological space M, let Rn.M/ be the space of SL.n; C/ representa- tions of π1.M/, and let Xn.M/ be the associated character variety. We use ξα to denote the character of the representation α V π1.M/ ! SL.n; C/. We will often make use of the important fact that two irreducible representations determine the same character if and only if they are conjugate; see [Lubotzky and Magid 1985, Corollary 1.33]. Now suppose K is a knot. The group Z=n has an action on the representation k 2πik=n variety Rn.NK /, given by twisting by the n-th roots of unity ! D e 2 U.1/. MSC2000: primary 57M25; secondary 20C15. Keywords: metabelian representation, knot group, character variety, group action, fixed point. 1 2 HANSU.BODEN AND STEFAN FRIEDL (This is a special case of the more general twisting operation described in [Lubotzky and Magid 1985, Chapter 5].) More precisely, we write Z=n D hσ j σ n D 1i and ".g/ set (σ ·α/.g/ D ! α.g/ for each g 2 π1.NK /, where " V π1.NK / ! H1.NK / D Z is determined by the given orientation of the knot. This constructs an action of Z=n on Rn.NK / which, in turn, descends to an action on the character variety Xn.NK /. Our main result identifies the fixed points ∗ of Z=n in Xn.NK / — the irreducible characters — as those associated to metabelian representations. Theorem 1. The character ξα of an irreducible representation α V π1.NK / ! SL.n; C/ is fixed under the Z=n action if and only if α is metabelian. In proving this result, we will actually characterize the entire fixed point set Z=n Xn.NK / in terms of characters ξα of the metabelian representations α D α.n,χ/ described in Section 2.3 (see Theorem 4). When n D 2, it turns out that every metabelian SL.2; C/ representation is dihedral. For this case, Theorem 1 was first proved by F. Nagasato and Y. Yamaguchi [2008, Proposition 4.8]. As an application of Theorem 1, we prove a result about the twisted Alexander polynomials associated to metabelian representations. This result was first shown by C. Herald, P. Kirk and C. Livingston [2010] using completely different methods. Our approach is elementary and natural, and is explained in Section 3.2, where we apply it to give an answer to a question raised by Hirasawa and Murasugi [2009]. 2. The classification of metabelian representations of knot groups We recall some results from [Boden and Friedl 2008] regarding the classification of metabelian representations of knot groups. 2.1. Preliminaries. Given a group π, we write π .n/ for the n-th term of the derived series of π. These subgroups are defined inductively by setting π .0/ D π and π .iC1/ DTπ .i/; π .i/U. The group π is called metabelian if π .2/ D feg. Suppose V is a finite-dimensional vector space over C. A representation % V π ! Aut.V / is called metabelian if % factors through π/π .2/. The representation % is called reducible if there exists a proper subspace U ⊂ V invariant under %(γ / for all γ 2 π. Otherwise, % is called irreducible or simple. If % is the direct sum of simple representations, then % is called semisimple. Two representations %1 V π ! Aut.V / and %2 V π ! Aut.W/ are called isomor- −1 phic if there exists an isomorphism ' V V ! W such that ' ◦ %1.g/ ◦ ' D %2.g/ for all g 2 π. 2.2. Metabelian quotients of knot groups. Let K ⊂ 63 be a knot in an integral homology 3-sphere. Denote by NeK the infinite cyclic cover of NK corresponding METABELIAN SL.n; C/ REPRESENTATIONSOFKNOTGROUPS,II 3 ∼ .1/ to the abelianization π1.NK / ! H1.NK / D Z. Thus, π1.NeK / D π1.NK / and ±1 ∼ .1/ .2/ H1 NK I ZTt U D H1.NeK / D π1.NK / =π1.NK / : The ZTt±1U-module structure is given on the right hand side by tn · g VD µ−n g µn, where µ is a meridian of K . For a knot K , we set π VD π1.NK / and consider the short exact sequence 1 ! π .1//π .2/ ! π/π .2/ ! π/π .1/ ! 1: .1/ ∼ Since π/π D H1.NK / D Z, this sequence splits, and we get isomorphisms .2/ ∼ .1/ .1/ .2/ ∼ .1/ .2/ ∼ ±1 π/π D π/π n π /π D Z n π /π D Z n H1.NK I ZTt U/; g 7! (µ".g/; µ−".g/g/ 7! .".g/; µ−".g/g/; where the semidirect products are taken with respect to the Z actions defined by .1/ .2/ n ±1 letting n 2 Z act on π /π by conjugation by µ , and on H1.NK I ZTt U/ by multiplication by tn.
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