to appear in "Knots and Primes", proceedings of the JAMI conference, Johns Hopkins Univeersity 2004, Contemporary Mathematics .pdf
This short note proves that the outer automorphism group of a free group acts ergodicly on the SU(2)-character variety. .ps , .dvi , or .pdf .
This short note proves that the outer automorphism group of a free group acts ergodicly on the SU(2)-character variety. .ps , .dvi , or .pdf .
This paper, written jointly with Richard Wentworth, discusses an invariant of surface group representations, a real-valued function on Teichmueller space. We discuss criteria for this function to be proper, and apply this to the action of the mapping class group on deformation spaces of flat bundles over closed surfaces. .ps , .dvi , or .pdf .
This paper, written jointly with Francois Labourie and Gregory Margulis, gives necessary and sufficient conditions for properness of an affine deformation of a convex cocompact discrete subgroup of O(2,1). This criterion uses a real-valued function on the space of geodesic currents generalizing Margulis's signed marked Lorentzian length spectrum invariant. .ps , .dvi , or .pdf .
This paper gives an elementary proof that the SL(2,C)-character variety of a rank two free group is an affine 3-space. It also derives defining equations for the character variety for a rank three free group. .ps , .dvi , or .pdf .
This joint paper with George Stantchev is based on George's 2003 doctoral dissertation. It describes the dynamics of the modular group action on characters of isometric actions of rank two free groups on hyperbolic 3-space which preserve a hyperbolic plane (but not necessarily an orientation on the hyperbolic plane). .ps , .dvi , or .pdf .
This paper deals with complex twist flows on complex character varieties and the deformation space of complex projective structures. .ps , .dvi , or .pdf . This paper appeared in the volume "Ergodic theory, geometry and arithmetic."
This paper deals with the dynamics of the action of the automorphism group of a the relative character varieties of a punctured torus. .ps , .dvi , or .pdf .
The main result of this paper, written jointly with Walter Neumann, is that the action of the modular group on the homology of the SL(2,C)-character varieties of a one-holed torus and a four-holed sphere factor through a finite group. .ps , .dvi , or .pdf . This paper has appeared in Mathematical Research Letters.
A preliminary version of a paper which appeared in Annals of Mathematics, Volume 146 (1997), pp.1--33. [ .ps or .dvi ]
Preliminary version of joint paper with Robert Benedetto, which appeared in Experimental Mathematics, (1999), 8 (1), pp.85--104. This research began as a "Research Experiences for Undergraduates" project in the summers of 1992 and 1993. [ .ps or .dvi ]
Lectures given at the workshop at Korea National University of Education in Ch'onju, Korea in 1994. [ .ps or .dv ]
This paper, The deformation theory of representations of fundamental groups of K\"ahler manifolds, coathored with John Millson, was published in Publications Mathematiques d'I. H. E. S., Vol.67 (1988), pp.43--96. We describe the local structure of the variety of representations of fundamental groups of compact Kaehler manifolds. The main result is that locally the space of representations is a cone defined by homogeneous quadratic equations in the tangent space. The proof uses the formality of a controlling differential graded Lie algebra, as suggested by P. Deligne.
This joint paper with Eugene Xia expounds the theory of Higgs line bundles over closed Riemann surfaces from the point of view of representations of the fundamental group. .ps , .dvi , or .pdf . This paper has recently been acceped for publication in the Memoirs of the American Mathematical Society.
Coauthored with Cathy Jones and Virginie Charette, this paper studies dynamical properties of geodesics in flat Lorentz space forms. Available in .dvi, .ps, or .pdf format.
This paper , coauthored with Todd Drumm, shows that the Margulis invariant is a complete invariant of the translational conjugacy class of an affine deformation of a Fuchsian group.
This paper , submitted to the Proceedings of the Newton Institute Euro-Workshop on ``Ergodic theory, rigidity and geometry'' surveys various properties of this invariant, interpreted in terms of deformations of hyperbolic structures on surfaces and the basic conjecture relating it to properness of affine deformations is discussed. In particular proper actions determine deformations of hyperbolic surfaces in which {\em all\/} the closed geodesics lengthen (or shorten). Formulas are derived showing that $\alpha$ grows linearly on along a coset of a hyperbolic one-parameter subgroup. An example of a deformation of hyperbolic surfaces is given along with the corresponding Margulis space-time.
In this paper, coauthored with Gregory Margulis, we prove the theorem, due to Geoff Mess, that the linear holonomy group of a complete flat Lorentz 3-manifold cannot be cocompact in SO(2,1). This paper appeared in the Proceedings of the Workshop ``Crystallographic Groups and their Generalizations II,'' which took place in Kortrijk, Belgium in May 1999 (Contemporary Mathematics 262, (2000), 135---146, American Mathematical Society).
An exposition, coauthored with Virginie Charette, of results based on the 1990 doctoral thesis of Todd Drumm. This paper appeared in the Proceedings of the Workshop ``Crystallographic Groups and their Generalizations II,'' which took place in Kortrijk, Belgium in May 1999 (Contemporary Mathematics 262, (2000), 69---98, American Mathematical Society).
A preliminary version of a survey paper, coauthored with Virginie Charette, Todd Drumm and Maria Morrill, which we will submit to the Proceedings of the Arthur Besse Table Ronde de Geometrie Pseudo-Riemannienne Globale, which took place at the Institut Elie Cartan in Nancy last June.
A revised version of a joint paper with Todd Drumm which appeared in International Journal of Mathematics, Volume 1 (1990), pp.149--161. [ .ps or .dvi ]
Preliminary version of a joint paper with Todd Drumm, which appeared in Topology 38 (2), (1999), pp. 323--351. [ .ps or .dvi ]
See also our Electronic Research Announcement summarizing these results.
Preliminary version of a survey article, which appeared in ``The Geometry of Group Representations,'' Contemp.\ Math.\ Vol.74, Amer.\ Math.\ Soc.\ (1988), 169--198 (W. Goldman and A. Magid, eds.).
This paper, written jointly with Oliver Baues, deals with the deformation space of complete affine structures on the 2-torus. A smooth structure is exhibited, using the periods of parallel exterior 1-forms, but it is shown that no universal smooth family exists. .ps , .dvi , or .pdf . This paper will appear in the Contemporary Mathematics volume, "Complex geometry and dynamical systems."a
Suhyoung Choi and I determine the deformation space of convex RP2-structures on all 2-dimensional orbifolds. Available in .pdf format.
Suhyoung Choi and I show that the deformation space of convex RP2-structures on all 2-dimensional orbifolds identifies with the "Teichmuller component" (as defined by Hitchin) in the PGL(3,R)-character variety. Available in .pdf format.
Written with Suhyoung Choi, a Research Report which recently appeared in the Bulletin of the American Mathematical Society.
Lecture notes of a course on affine and projective structures on manifolds, given at the University of Maryland in the Spring semester of 1988. [ .ps or .dvi ]
A preliminary version of a paper, coauthored with Mehdi-Reza Darvishzadeh, describing geometric structures on the deformation space of real projective structures. This paper appeared in Journal of the Korean Math. Soc. Vol~33 (1996), 625--638.
Lectures given at a workshop at the Topology and Geometry Research Center, Kyungpook National University in Taegu, Korea in 1994.
Excerpts (including pictures) of my book, Complex Hyperbolic Geometry which was recently published in the Oxford Mathematical Monographs, by Oxford University Press (xvii + 316 pp. + 58 illus. (1999) ISBN 0-19-853793-X).
A preliminary version of a joint paper with Misha Kapovich and Bernhard Leeb, describing new examples of discrete faithful actions of surface groups on complex hyperbolic two-space. This paper recently appeared in Communications in Geometry and Analysis (vol.\ 9, no.\ 1, January 2001, 61--96). [ .ps or .dvi ]
I am very grateful for support of my research by various organizations.
I also review papers and books for Mathematical Reviews, Zentralblatt fur Mathematik, the Bulletin of the American Mathematical Society, the American Mathematical Monthly and SIAM Review. Here are some of my longer reviews:
Autos.nb A Mathematica notebook for investigating automorphisms of free groups and their SL(2)-character varieties.
Rep.nb A Mathematica notebook for investigating representative functions on the SL(2)-character variety of a rank two free group.
AffineT2.nb A Mathematica notebook for drawing developing maps of complete affine structures on the 2-torus.
Cubic1.nb A Mathematica notebook for drawing cubic surfaces arising as moduli spaces for SL(2)-characters of a one-holed torus.