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Bounded cohomology of subgroups of mapping class groups
 Geom. Topol. 6 (2002) 69–89. MR1914565 (2003f:57003), Zbl 1021.57001
"... groups ..."
Growth of maps, distortion in groups and symplectic geometry
, 2000
"... In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: • The uniform norm of the differential of its nth iteration; • The word length of its nth iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffe ..."
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Cited by 23 (0 self)
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In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: • The uniform norm of the differential of its nth iteration; • The word length of its nth iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove nonexistence of certain nonlinear symplectic representations for finitely generated groups.
Curvature and rank of Teichmüller space
 Amer. J. Math
, 2001
"... Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the WeilPetersson metric on Teichmüller space Teich(S) is Gromovhyperb ..."
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Cited by 21 (1 self)
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Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the WeilPetersson metric on Teichmüller space Teich(S) is Gromovhyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3 the WeilPetersson metric has higher rank in the sense of Gromov (it admits a quasiisometric embedding of R k, k ≥ 2); when d(S) ≤ 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of CP(S) prove Gromovhyperbolicity. We prove moreover that Teich(S) admits no geodesically complete Gromovhyperbolic metric of finite covolume when d(S) ≥ 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) ≥ 2. 1
WeilPetersson completion of Teichmüller spaces and mapping class group actions, preprint math.DG/0112001
"... Given a surface of higher genus, we will look at the WeilPetersson completion of the Teichmüller space of the surface, and will study the isometric action of the mapping class group on it. The main observation is that the geometric characteristics of the setting bear strong similarities to the ones ..."
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Cited by 15 (2 self)
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Given a surface of higher genus, we will look at the WeilPetersson completion of the Teichmüller space of the surface, and will study the isometric action of the mapping class group on it. The main observation is that the geometric characteristics of the setting bear strong similarities to the ones in semisimple Lie group actions on noncompact symmetric spaces. 1
Automorphism groups of free groups, surface groups and free abelian groups. Problems on mapping class groups and related topics
 301¯D316, Proc. Sympos. Pure Math
, 2006
"... The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2dimensional torus. Thus this group is the beginning of three natural sequences of ..."
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Cited by 12 (0 self)
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The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the mapping class groups Mod ± (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a successful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similarities and differences between these series of groups and present some open problems motivated by this philosophy.
Area preserving group actions on surfaces
"... Suppose G is an almost simple group containing a subgroup isomorphic to the threedimensional Heisenberg group. For example any finite index subgroup of SL(3, Z) is such a group. The main result of this paper is that every action of G on a closed oriented surface by area preserving diffeomorphisms f ..."
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Cited by 8 (4 self)
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Suppose G is an almost simple group containing a subgroup isomorphic to the threedimensional Heisenberg group. For example any finite index subgroup of SL(3, Z) is such a group. The main result of this paper is that every action of G on a closed oriented surface by area preserving diffeomorphisms factors through a finite group. 1
Computing the shortest essential cycle
, 2008
"... An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of ..."
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Cited by 8 (4 self)
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An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of boundaries are fixed. Our result corrects an error in a paper of Erickson and HarPeled.
The Torelli geometry and its applications: research announcement
"... Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod ± (S) of S which is defined as the group of isotopy classes of all di ..."
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Cited by 7 (0 self)
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Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod ± (S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skewsymmetric, bilinear form on H = H1(S, Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g, Z) and the action of ModS on H leads to a natural surjective homomorphism Mod(S) − → Sp(2g, Z). The Torelli group of S, denoted by IS, is defined as the kernel of this homomorphism; that is, IS is the subgroup of Mod(S) consisting of elements which act trivially on H1(S, Z). In particular, we have the following wellknown exact sequence
ENERGY OF TWISTED HARMONIC MAPS OF RIEMANN SURFACES
, 2004
"... The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over Riemann surfaces is a function on Teichmüller space which is a qualitative invariant of the holonomy representation. Adapting ideas of SacksUhlenbeck, SchoenYau and Tromba, we show that the energy fu ..."
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Cited by 3 (2 self)
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The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over Riemann surfaces is a function on Teichmüller space which is a qualitative invariant of the holonomy representation. Adapting ideas of SacksUhlenbeck, SchoenYau and Tromba, we show that the energy function is proper for any convex cocompact representation of the fundamental group. As a corollary, the mapping class group acts properly on the subset of convex cocompact representations.