On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds

We apply the concept of asymptotic cone to distinguish quasi-isometry classes of fundamental groups of 3-manifolds. We prove that the existence of a Seifert component in a Haken manifold is a quasi-isometry invariant of its fundamental group.This research was partially supported by the grant SFB 256 Nichtlineare partielle Differentialgleichungen and the NSF grant DMS-9306140 (Kapovich).     

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.     

Subset currents on free groups

We introduce and study the space ${{\mathcal{S}{\rm Curr} (F_N)}}$ of subset currents on the free group F _N , and, more generally, on a word-hyperbolic group. A subset current on F _N is a positive F _N -invariant locally finite Borel measure on the space ${{\mathfrak{C}_N}}$ of all closed subsets of ?F _N consisting of at least two points. The well-studied space Curr(F _N ) of geodesics currents–positive F _N -invariant locally finite Borel measures defined on pairs of different boundary points–is contained in the space of subset currents as a closed ${{\mathbb{R}}}$ -linear Out(F _N )-invariant subspace. Much of the theory of Curr(F _N ) naturally extends to the ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ context, but new dynamical, geometric and algebraic features also arise there. While geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in F _N . If a free basis A is fixed in F _N , subset currents may be viewed as F _N -invariant measures on a “branching” analog of the geodesic flow space for F _N , whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of F _N with respect to A. Similarly to the case of geodesics currents, there is a continuous Out(F _N )-invariant “co-volume form” between the Outer space cv _N and the space ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ of subset currents. Given a tree ${{T \in {\rm cv}_N}}$ and the “counting current” ${{\eta_H \in \mathcal{S}\;{\rm Curr} (F_N)}}$ corresponding to a finitely generated nontrivial subgroup H ≤  F _N , the value ${{\langle T, \eta_H \rangle}}$ of this intersection form turns out to be equal to the co-volume of H, that is the volume of the metric graph T _H /H, where ${{T_H \subseteq T}}$ is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form ${{{\rm cv}_N \times \mathcal{S}\;{\rm Curr}(F_N)\; \to [0,\infty)}}$ does not extend to a continuous map ${{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}$ .     

Conformally flat structures on 3-manifolds: Existence problem

Dedicated to Yurii Grigor'evich Reshetnyak on his sixtieth birthday.     

On monodromy of complex projective structures

Summary We prove that for any nonelementary representation : ₁(S SL (2, )) of the fundamental group of a closed orientable hyperbolic surfaceS there exists a complex projective structure onS with the monodromy .Oblatum IV-1993 & 24-IV-1994     

Small cancellation groups and translation numbers

In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any and any there is an algorithm deciding whether or not the equation has a solution. There is also an algorithm for calculating for each the maximum such that is an -th power of some element. We also note that these groups cannot contain isomorphic copies of the group of -adic fractions and so in particular of the group of rational numbers. Besides we show that for and groups all translation numbers are rational and have bounded denominators.

     

Quasi-isometries preserve the geometric decomposition of Haken manifolds

We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic. We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by-product we describe all 2-dimensional quasi-flats in the universal covers of non-geometric Haken manifolds. Oblatum 27-III-1996 & 5-IX-1996