K_g is not finitely generated
We prove that for any genus g>1, the subgroup K_g of the mapping class group of a closed genus g surface generated by Dehn twists about separating curves is not finitely generated.
math.GT↗
arXiv subjects
Benson Farb
Publications and source records attributed to Benson Farb.
Commensurability invariants for nonuniform tree lattices
We study nonuniform lattices in the automorphism group G of a locally finite simplicial tree X. In particular, we are interested in classifying lattices up to commensurability in G. We introduce two new commensurability invariants: quotient growth, which measures the growth of the noncompact quotient of the lattice; and stabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. When X is the biregular tree X_{m,n}, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real number νwe construct uncountably many noncommensurable lattices with covolume ν.
math.GR↗
Hidden symmetry and arithmetic manifolds
A "hidden symmetry" of a Riemannian manifold M is an isometry of a d-sheeted, 1<d<\infty, Riemannian cover of M which is not the lift of any isometry. In this paper we characterize the locally symmetric metric(s) on a closed, arithmetic manifold as the unique metric with infinitely many hidden symmetries.
math.DG↗
Every mapping class group is generated by 6 involutions
Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements suffice to generate Mod_{g,0}. We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod_{g,b} but also for the order of those elements. In particular, our main result is that 6 involutions (i.e. orientation-preserving diffeomorphisms of order two) suffice to generate Mod_{g,b} for every genus g >= 3, b = 0, and g >= 4, b = 1.
math.GT↗
The Torelli geometry and its applications
For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometry is induced by a diffeomorphism of the surface in question. We also provide an intrinsic algebraic characterization of some natural elements of the Torelli group and of some geometric relations between them. When combined, these results allow us to compute the automorphism group, the outer automorphism group and the abstract commensurator of the Torelli group for surfaces of genus at least 5.
math.GT↗
Convex cocompact subgroups of mapping class groups
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 product of pi_1(S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G=Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.
math.GR↗
Some recent applications of the barycenter method in geometry
In this paper we describe some recent applications of the barycenter method in geometry. This method was first used by Duady-Earle and later greatly extended by Besson-Courtois-Gallot in their solution of a number of long-standing problems, in particular in their proof of entropy rigidity for closed, negatively curved locally symmetric manifolds. Since there are already a number of surveys describing this work, we will concentrate here only on advances that have occured after these surveys appeared. While most of this paper is a report on results appearing in other papers, some of the material here is new.
math.DG↗
Real-analytic, volume-preserving actions of lattices on 4-manifolds
We prove that any real-analytic, volume-preserving action of a lattice $Γ$ in a simple Lie group with $\Qrank(Γ)\geq 7$ on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.
math.DG↗
Curvature and rank of Teichmüller space
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 Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) <= 2. When d(S) >= 3 the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric 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 Gromov-hyperbolicity. We prove moreover that Teich(S) admits no geodesically complete Gromov-hyperbolic 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.
math.GT↗
Groups of homeomorphisms of one-manifolds, I: actions of nonlinear groups
This self-contained paper is part of a series \cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1) Any homomorphism of (almost any) mapping class group or automorphism group of a free group into $\Diff_+^r(S^1), r\geq 2$ is trivial. For r=0 Nielsen showed that in many cases nontrivial (even faithful) representations exist. Somewhat weaker results are proven for finite index subgroups. 2) We construct a finitely-presented group of real-analytic diffeomorphisms of $\R$ which is not residually finite.
math.DS↗
Group actions on one-manifolds, II: Extensions of Hölder's Theorem
We study groups of homeomorphisms of R, each of whose elements have at most one fixed point. In particular we prove that any such group of C^2 diffeomorphisms is topologically conjugate to an affine group.
math.DS↗
Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups
This paper has been withdrawn by the authors; it will be incorporated into part I of the series (in preparation).
math.DS↗
Problems on the geometry of finitely generated solvable groups
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.
math.GR↗
A strong Schottky Lemma for nonpositively curved singular spaces
In this paper we give a criterion for pairs of isometries of a nonpositively curved metric space to generate a free group. This criterion holds only in singular spaces, for example in Euclidean buildings. The original motivation for our criterion was to prove that the four dimensional Burau representation is faithful. Although we do not settle this question, we do exhibit a related 2-parameter family of faithful representations of the free group F_2.
math.GR↗
The Degree Theorem in higher rank
Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive curvature. Assume that M has no locally Euclidean factors and no factors locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any continuous map f:N -> M, we show there is a C^1 representative in the homotopy class of f with Jacobian bounded by a universal constant C depending only on n and the smallest Ricci curvatures of N and M. This implies that deg(f)<= C Vol(N)/Vol(M). For M negatively curved this was proved by Gromov. Two corollaries of our result are that Minvol(N)>0 whenever deg(f)<>0 and a simple proof of G. Prasad's result that lattices in semi-simple Lie groups are co-Hopfian. We prove a version of all results for finite volume manifolds as well.
math.DG↗
Minimal entropy rigidity for lattices in products of rank one symmetric spaces
We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as well.
math.DG↗
The geometry of surface-by-free groups
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 example of how quasi-isometries can actually detect finite index information. The proofs of these theorems involve ideas from coarse topology, Teichmuller geometry, pseudo-Anosov dynamics, and singular solv-geometry.
math.GR↗
On the asymptotic geometry of abelian-by-cyclic groups
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 such that M^r, N^s have the same absolute Jordan form. We also prove quasi-isometric rigidity: if Gamma_M is an abelian-by-cyclic group determined by an n x n integer matrix M with |det(M)| > 1, and if G is any finitely generated group quasi-isometric to Gamma_M, then there is a finite normal subgroup K of G such that G/K is abstractly commensurable to Gamma_N, for some n x n integer matrix N with |det(N)| > 1.
math.GR↗