Rank-1 phenomena for mapping class groups

By Benson Farb, Alexander Lubotzky, Yair N. Minsky. 2001

112 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b2e5cfe4e902595d7b80a591ae3c3130d1a17543

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Citations in the corpus

Curvature and rank of Teichmüller space (2001, background): “2], but bounding the rank from above appears delicate. Remark. While Dehn twists about disjoint simple closed curves produce quasi-ﬂats in the mapping class group (endowed with the word metric) - see [FLM], Dehn twist orbits in Teich(S) do not generate quasi-ﬂats in either the Weil-Petersson or Teic”
Dimension and rank for mapping class groups (2005, background): “Moreover, such subgroups are known to be quasi-isometrically embedded by results in [Mos], when S has punctures, and by [ FLM ] in the general case.”
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity (2005, background): “roup of MCG(S). Moreover, it THICK METRIC SPACES, RELATIVE HYPERBOLICITY, & QUASI-ISOMETRIC RIGIDITY 15 has been shown that these Abelian subgroups are quasi-isometrically embedded in MCG(S) (see [FLM] and [Mos]). Masur and Minsky showed that MCG(S) is weakly relatively hyperbolic with respect t”
Coarse median spaces and groups (2013, background): “One can show that some (in fact any) free abelian subgroup of Map(6) of rank ξ(6) is necessarily quasi-isometrically embedded [Farb et al. 2001].”
Constructing group actions on quasi-trees and applications to mapping class groups (2010, background): “ving unbounded orbits. See Theorem 5.9. In the case of odd genus one has to pass to a subgroup of ﬁnite index. It follows that each Dehn twist has linear growth in the word length in MCG(Σ) (known by [FLM01]). Theorem F provides a sharp contrast to a result of Bridson [Bri10], who showed that in sem”
Divergence in lattices in semisimple Lie groups and graphs of groups (2008, result): “r irreducible non-uniform lattices this is so because such lattices contain distorted cyclic subgroups[LMR93], while the mapping class groups do not, according to results of Farb, Lubotzky and Minsky [FLM01]. On the other hand, uniform lattices do not have cut-points in their asymptotic cones (see a”
Geometry and rigidity of mapping class groups (2008, result): “The list of such results is intriguing but incomplete: see for example McCarthy [40] and Ivanov [30] (a Tits-alternative), Ivanov [32] and Farb-Masur [17] (non-arithmeticity and superrigidity), Harvey [28] (analogy to Bruhat-Tits buildings), Harer [27] (homology stability), Farb-Lubotzky-Minsky [16]”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, unknown): “In MCGpSq, the subgroup xταy generated by the Dehn twist about α is quasiisometrically embedded [FLM01], but in pT pSq, dT q, the orbit of τα is distorted.”
Asymptotic Geometry of the Mapping Class Group and Teichmuller Space (2005, methodology): “ehn twists along a maximal collection of simple closed curves yield a quasi-isometric embedding of R3g+p−3 into M. This fact was ﬁrst proven for punctured surfaces in [Mo], then in full generality in [FLM]. We remark that it also follows directly by applying Theorem 3.18 (the analogous argument to p”
Convex cocompact subgroups of mapping class groups (2001, background): “Masur and Minsky [33] give an example of an infinite cyclic subgroup of Isom(T ) which is not orbit quasiconvex, and yet this subgroup is quasi-isometrically embedded in MCG [14]; it may also be quasiconvex in MCG, but we have not investigated this.”
Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (2010, background): “It is well-known that mapping class groups share many properties with lattices in rank one semisimple Lie groups as well as properties of lattices in higher rank semisimple Lie groups (see the paper [23] of Farb, Lubotzky and Minsky, for instance).”
The Mapping Class Group from the Viewpoint of Measure Equivalence Theory (2005, unknown): “, see [20], [47], [50]).”
Some problems on mapping class groups and moduli space (2006, background): “ely related to, but diﬀerent than, the question of convex cocompactness of subgroups of Mod g, as deﬁned in [FMo], since the embedding of Mod g in Teich g via any orbit is exponentially distorted, by [FLMi]. 3.3 Decision problems for subgroups As a collection of groups, how rich and varied can the s”
Growth of maps, distortion in groups and symplectic geometry (2001, methodology): “David Fisher pointed out that the argument based on [ FLM ] yields the statement of the corollary for lattices in some more general locally compact groups.”
Fifteen problems about the mapping class groups (2006, background): “hmic growth corresponds tovirtually unipotent elements of arithmetic groups of rank ≥ 2, according to a theorem of A. Lubotzky, S. Moses and M.S. Raghunathan [LMR]. B. Farb, A. Lubotzky and Y. Minsky [FLM] proved that the growth is linear, so the problem is solved completely. 10. Nonorientable Surfa”