We prove the following:
1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+ε) bi-Lipschitz homeomorphism between S'_1 and S'_2 and both covers S'_i have bounded injectivity radius.
2. Let M be a closed hyperbolic 3-manifold. Then there exists a map j: S -> M where S is a surface of bounded injectivity radius and j is a pi_1-injective local isometry onto its image.
We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces (the good pants are pairs of pants whose cuffs have the length nearly equal to some large number R). Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpreis conjecture.
Let \hat{S} be the algebraic universal cover of a closed surface of genus >1, T(\hat{S}) its Teichmuller space, M(\hat{S}) the group of mapping classes stabilizing a fixed leaf l. The L^1 Ehrenpreis conjecture asserts that M(\hat{S}) on T(\hat{S}) with dense orbits with respect to the L^1 topology (the topology induced by the L^1 norm on quadratic differentials). We give a proof of this weaker form of the Ehrenpreis conjecture.
We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly map for such groups is an isomorphism.
We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.
The Hanna Neumann conjecture gives a bound on the intersection of finitely generated subgroups of free groups. We explore a natural extension of this result, which turns out to be true only in the finite index case, and provide counterexamples for the general case. We also see that the graph-based method of generating random subgroups of free groups developed by Bassino, Nicaud and Weil is well-suited to generating subgroups with non-trivial intersections. The same method is then used to generate a counterexample to a similar conjecture of Guzman.
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of …
A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counterexample to this conjecture, then there must be a counterexample of a particularly simple form. We prove these results by relating the conjecture to a family of linear representations of the mapping class group that we call the higher Prym representations. They generalize the cla…
Let $\mathcal{M}_{g,ε}$ be the $ε$-thick part of the moduli space $\mathcal{M}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover $\mathcal{M}_{g,ε}$ is bounded below by $(c_1g)^{2g}$ and bounded above by $(c_2g)^{2g}$, where the constants $c_1,c_2$ depend only on $ε$ and $r$, and in particular not on $g$. Using the counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichmüller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates…
For some $g \geq 3$, let $Γ$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Γ$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n \geq 1$, we have $T_x^n \in Γ$. We prove that $T_x^n$ is torsion in the abelianization of $Γ$. Our second result shows that the abelianization of $Γ$ is finite if $Γ$ contains a "large chunk" (in a certain…
We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.
Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学）；Institut Universitaire de France（法国高等研究院）；Department of Mathematics, Michigan State University（密歇根州立大学数学系）
Around twenty years ago Ghys conjectured that finite subgroups of the diffeomorphism group of a compact smooth manifold M have an abelian normal subgroup of index at most a(M), where a(M) depends only on M. First we construct a family of counterexamples to this conjecture including, for example, the product space $T^2\times S^2$. Following the first appearance of our counterexample on the arXiv Ghys put forward a revised conjecture, which predicts only the existence of a nilpotent normal subgroup of index at most n(M). Our main result is the proof of the revised Ghys conjecture. More generally…
We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being non-empty implies that the intersection of the subgroups is non-trivial. We present examples to show that a conjecture of Susskind, stating that the intersection of the sets of conical limit points of subgroups $Φ$ and $Θ$ of a Kleinian group $Γ$ is contained in the limit set of …
Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove that there exists a division ring D(G) such that A[G] < D(G) < U(G). This establishes some versions of the Atiyah conjecture for the group G.
For a certain class of compact oriented 3-manifolds, M. Goussarov and K. Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ``cut-and-paste'' operations defined by the lower central series of the Torelli group of a surface. In this paper, we observe that this is a variation of a classical problem in group theory, namely the ``dimension subgroup problem.'' This viewpoint allows us to prove, by purely algebraic methods, an analogue of the Goussarov-Habiro conjecture for finite-type invariants with values in a fixed field. We deduce …
We show that for groups acting acylindrically on simplicial trees the $K$- and $L$-theoretic Farrell-Jones Conjecture relative to the family of subgroups consisting of virtually cyclic subgroups and all subconjugates of vertex stabilisers holds. As an application, for amalgamated free products acting acylindrically on their Bass-Serre trees we obtain an identification of the associated Waldhausen Nil-groups with a direct sum of Nil-groups associated to certain virtually cyclic groups. This identification generalizes a result by Lafont and Ortiz. For a regular ring and a strictly acylindrical a…
The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of $G$ are separated by a quasi-convex surface subgroup. Thus, the Cannon's conjecture is reduced to showing that such a group contai…
Let N be a closed, oriented 3-manifold. A folklore conjecture states that $S^{1} \times N$ admits a symplectic structure if and only if $N$ admits a fibration over the circle. We will prove this conjecture in the case when N is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes 3-manifolds with vanishing Thurston norm, graph manifolds and 3-manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic 3-manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots…
We identify a universal group $U$ and show that $\Bbb H^3/G$ is $S^3$ when $G$ is a finite index subgroup of $U$ generated by elements of finite order.
