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Sep 22, 2013
09/13

by
Lee Mosher

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This document is a practical guide to computations using an automatic structure for the mapping class group of a once-punctured, oriented surface $S$. We describe a quadratic time algorithm for the word problem in this group, which can be implemented efficiently with pencil and paper. The input of the algorithm is a word, consisting of ``chord diagrams'' of ideal triangulations and elementary moves, which represents an element of the mapping class group. The output is a word called a ``normal...

Source: http://arxiv.org/abs/math/9409209v1

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Sep 22, 2013
09/13

by
Lee Mosher

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In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for the mapping class group MCG(S) of a once punctured surface S of genus at least 2: if K is a finitely generated group quasi-isometric to MCG(S) then there is a homomorphism K -> MCG(S) with finite kernel and finite index image. This theorem is joint with Kevin Whyte.

Source: http://arxiv.org/abs/math/0308065v1

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Sep 22, 2013
09/13

by
Lee Mosher

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The theme of this survey is that subgroups of the mapping class group of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of the Teichmuller space of S, just as discrete subgroups of the isometries of hyperbolic space can be studied via their action on compactified hyperbolic space.

Source: http://arxiv.org/abs/math/0702428v1

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Sep 22, 2013
09/13

by
Lee Mosher

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Let S be a closed, oriented surface of genus at least 2, and consider the extension 1 -> pi_1 S -> MCG(S,p) -> MCG(S) -> 1, where MCG(S) is the mapping class group of S, and MCG(S,p) is the mapping class group of S punctured at p. We prove that any quasi-isometry of MCG(S,p) which coarsely respects the cosets of the normal subgroup pi_1 S is a bounded distance from the left action of some element of MCG(S,p). Combined with recent work of Kevin Whyte this implies that if K is a...

Source: http://arxiv.org/abs/math/0308067v1

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Jul 20, 2013
07/13

by
Lee Mosher

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We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over gamma is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental...

Source: http://arxiv.org/abs/math/0107035v3

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Sep 21, 2013
09/13

by
Lee Mosher

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The quotient of a biautomatic group by a subgroup of the center is shown to be biautomatic. The main tool used is the Neumann-Shapiro triangulation of $S^{n-1}$, associated to a biautomatic structure on ${\Bbb Z}^n$. As an application, direct factors of biautomatic groups are shown to be biautomatic.

Source: http://arxiv.org/abs/math/9404203v1

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Sep 18, 2013
09/13

by
Benson Farb; Lee Mosher

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A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only if |det(M)|=1. We give a complete classification of the nonpolycyclic groups Gamma_M up to quasi-isometry: given n x n integer matrices M,N with |det(M)|, |det(N)| > 1, the groups Gamma_M, Gamma_N are quasi-isometric if and only if there exist positive integers r,s...

Source: http://arxiv.org/abs/math/0005181v1

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Sep 23, 2013
09/13

by
Michael Handel; Lee Mosher

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We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.

Source: http://arxiv.org/abs/1111.1994v2

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Sep 20, 2013
09/13

by
Michael Handel; Lee Mosher

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A fully irreducible outer automorphism phi of the free group F_n of rank n has an expansion factor which often differs from the expansion factor of the inverse of phi. Nevertheless, we prove that the ratio between the logarithms of the expansion factors of phi and its inverse is bounded above by a constant depending only on the rank n. We also prove a more general theorem applying to an arbitrary outer automorphism of F_n and its inverse, and their entire spectrum of expansion factors.

Source: http://arxiv.org/abs/math/0410015v2

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Sep 18, 2013
09/13

by
Benson Farb; Lee Mosher

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We show that every word hyperbolic, surface-by-(noncyclic) free group Gamma is as rigid as possible: the quasi-isometry group of Gamma equals the abstract commensurator group Comm(Gamma), which in turn contains Gamma as a finite index subgroup. As a corollary, two such groups are quasi-isometric if and only if they are commensurable, and any finitely generated group quasi-isometric to Gamma must be weakly commensurable with Gamma. We use quasi-isometries to compute Comm(Gamma) explicitly, an...

Source: http://arxiv.org/abs/math/0008215v1

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Sep 22, 2013
09/13

by
Michael Handel; Lee Mosher

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For any subgroup H of Out(F_n), either H has a finite index subgroup that fixes the conjugacy class of some proper, nontrivial free factor of F_n, or H contains a fully irreducible element phi, meaning that no positive power of phi fixes the conjugacy class of any proper, nontrivial free factor of F_n.

Source: http://arxiv.org/abs/0908.1255v1

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Sep 22, 2013
09/13

by
Michael Handel; Lee Mosher

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This is the introduction to a series of four papers that develop a decomposition theory for subgroups of Out(F_n) which generalizes the theory for elements of Out(F_n) found in the work of Bestvina, Feighn, and Handel on the Tits alternative, and which is analogous to the decomposition theory for subgroups of mapping class groups found in work of Ivanov. In this introduction we state the main theorems and we outline the contents of the whole series.

Source: http://arxiv.org/abs/1302.2681v2

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Sep 22, 2013
09/13

by
Michael Handel; Lee Mosher

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This is the second in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we relativize the "Kolchin-type theorem" from the work of Bestvina, Feighn, and Handel on the Tits alternative, which describes a decomposition theory for subgroups H of Out(F_n) all of whose elements have polynomial growth. The Relative Kolchin Theorem allows subgroups H whose elements have exponential...

Source: http://arxiv.org/abs/1302.2379v3

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Sep 18, 2013
09/13

by
Michael Handel; Lee Mosher

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We develop a notion of axis in the Culler--Vogtmann outer space X_r of a finite rank free group F_r, with respect to the action of a nongeometric, fully irreducible outer automorphism phi. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmuller space, X_r has no natural metric, and phi seems not to have a single natural axis. Instead our axes for phi, while not unique, fit into an ``axis bundle'' A_phi with nice...

Source: http://arxiv.org/abs/math/0605355v2

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Sep 19, 2013
09/13

by
Michael Handel; Lee Mosher

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Given a free factor A of the rank n free group F_n, we characterize when the subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1}) in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) of the conjugacy class of any free splitting of F_n is nondistorted, and we characterize when the stabilizer of the conjugacy class of an arbitrary free factor system of F_n is distorted. In all proofs...

Source: http://arxiv.org/abs/1009.5018v3

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Sep 20, 2013
09/13

by
Michael Handel; Lee Mosher

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We study those fully irreducible outer automorphisms phi of a finite rank free group F_r which are ``parageometric'', meaning that the attracting fixed point of phi in the boundary of outer space is a geometric R-tree with respect to the action of F_r, but phi itself is not a geometric outer automorphism in that it is not represented by a homemorphism of a surface. Our main result shows that the expansion factor of phi is strictly larger than the expansion factor of the inverse of phi. As...

Source: http://arxiv.org/abs/math/0410018v2

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Sep 22, 2013
09/13

by
Michael Handel; Lee Mosher

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This is the first in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we develop further the theory of geometric EG strata of relative train track maps originally introduced in the work of Bestvina, Feighn, and Handel on the Tits alternative, with our focus trained on certain 2-dimensional models of such strata called "geometric models" and on the interesting properties of...

Source: http://arxiv.org/abs/1302.2378v3

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Sep 22, 2013
09/13

by
Benson Farb; Lee Mosher

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We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a "Schottky subgroup" of MCG, the converse is true as well; a semidirect...

Source: http://arxiv.org/abs/math/0106190v3

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Sep 18, 2013
09/13

by
Benson Farb; Lee Mosher

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Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

Source: http://arxiv.org/abs/math/9809010v1

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Sep 18, 2013
09/13

by
Benson Farb; Lee Mosher

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A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.

Source: http://arxiv.org/abs/math/0005184v2

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Jun 30, 2018
06/18

by
Michael Handel; Lee Mosher

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We study the loxodromic elements for the action of $Out(F_n)$ on the free splitting complex of the rank $n$ free group $F_n$. We prove that each outer automorphism is either loxodromic, or has bounded orbits without any periodic point, or has a periodic point; and we prove that all three possibilities can occur. We also prove that two loxodromic elements are either co-axial or independent, meaning that their attracting/repelling fixed point pairs on the Gromov boundary of the free splitting...

Topics: Mathematics, Group Theory

Source: http://arxiv.org/abs/1402.1886

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Sep 20, 2013
09/13

by
Lee Mosher; Michah Sageev; Kevin Whyte

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This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincare duality groups. The main theorem says that, under certain hypotheses, if G is a finite graph of coarse Poincare duality groups then any finitely generated group quasi-isometric...

Source: http://arxiv.org/abs/math/0405237v2

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Sep 18, 2013
09/13

by
Lee Mosher; Michah Sageev; Kevin Whyte

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We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The second theorem gives general constructions for maximally symmetric trees, showing for instance that...

Source: http://arxiv.org/abs/math/0012004v2

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Sep 19, 2013
09/13

by
Lee Mosher; Michah Sageev; Kevin Whyte

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Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's Theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact...

Source: http://arxiv.org/abs/math/0010136v2

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Sep 18, 2013
09/13

by
Lee Mosher; Michah Sageev; Kevin Whyte

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We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For example, if Gamma is a graph of coarse PD(n) groups for a fixed n, if the Bass-Serre tree of Gamma has infinitely many ends, and if H is a finitely generated group quasi-isometric to pi_1(Gamma), then we prove that H is the fundamental group of a graph of coarse...

Source: http://arxiv.org/abs/math/0005210v1

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Sep 22, 2013
09/13

by
Howard Masur; Lee Mosher; Saul Schleimer

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We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic. As another application we show that train track sliding and splitting sequences give quasi-geodesics in the train track graph, generalizing a result of Hamenstaedt [Invent....

Source: http://arxiv.org/abs/1004.4564v1

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Sep 23, 2013
09/13

by
Jason Behrstock; Bruce Kleiner; Yair Minsky; Lee Mosher

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We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several...

Source: http://arxiv.org/abs/0801.2006v4