Acylindrical accessibility for groups

We define the notion of acylindrical graph of groups of a group. We bound the combinatorics of these graphs of groups for f.g. freely indecomposable groups. Our arguments imply the finiteness of acylindrical surfaces in closed 3-manifolds [Ha], finiteness of isomorphism classes of small splittings of (torsion-free) freely indecomposable hyperbolic groups as well as finiteness results for small splittings of f.g. Kleinian and semisimple discrete groups acting on non-positively curved simply connected manifolds. In order to get our accessibility for f.g. groups we generalize parts of Rips' analysis of stable actions of f.p. groups on real trees to f.g. groups. The concepts we present play an essential role in constructing the canonical JSJ decomposition ([Se1],[Ri-Se2]), in obtaining the Hopf property for hyperbolic groups [Se2], and in our study of sets of solutions to equations in a free group [Se3]. Oblatum 30-IV-1992 & 1-X 1996     

The isomorphism problem for toral relatively hyperbolic groups

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

A Remark on Twists and the Notion of Torsion-free Discrete Quantum Groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki–Takai duality. The twisted Takesaki–Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld–Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.     

Canonical representatives and equations in hyperbolic groups

Summary We use canonical representatives in hyperbolic groups to reduce the theory of equations in (torsion-free) hyperbolic groups to the theory in free groups. As a result we get an effective procedure to decide if a system of equations in such groups has a solution. For free groups, this question was solved by Makanin [Ma]|and Razborov [Ra]. The case of quadratic equations in hyperbolic groups has already been solved by Lysenok [Ly]. Our whole construction plays an essential role in the solution of the isomorphism problem for (torsion-free) hyperbolic groups ([Se1],[Se2]).Oblatum 1-1992 & 1-XI-1994Partially supported by NSF grant DMS-9305848     

二秩无扭群的自同构群和只有两个自同构的二秩无扭群

本文利用Ｋｕｒǒｓ不变量理论和不定方程理论，讨论了二秩无扭群的自同构群，以及有零高元的二秩无扭群只有两个自同构的充要条件．     

Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

Thick metric spaces,relative hyperbolicity,and quasi-isometric rigidity

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.     

Novikov Conjectures for Arithmetic Groups with Large Actions at Infinity

We construct a new compactification of a noncompact rank one globally symmetric space. The result is a nonmetrizable space which also compactifies the Borel–Serre enlargement X of X, contractible only in the appropriate ech sense, and with the action of any arithmetic subgroup of the isometry group of X on X not being small at infinity. Nevertheless, we show that such a compactification can be used in the approach to Novikov conjectures developed recently by G. Carlsson and E. K. Pedersen. In particular, we study the nontrivial instance of the phenomenon of bounded saturation in the boundary of X and deduce that integral assembly maps split in the case of a torsion-free arithmetic subgroup of a semi-simple algebraic Q-group of real rank one or, in fact, the fundamental group of any pinched hyperbolic manifold. Using a similar construction we also split assembly maps for neat subgroups of Hilbert modular groups.     

Quotient Divisible Groups of Rank 2

Mathematical Notes - In the paper, representations of torsion-free Abelian groups of rank $$2$$ using torsion-free groups of rank $$1$$ are studied. Necessary and sufficient conditions are found...     

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.      

Abelian Groups and Regular Modules

The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.     

Boundaries and JSJ Decompositions of CAT(0)-Groups

Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ?X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJ decomposition from ?X. We show that any discrete action of a group G on a CAT(0) space X satisfies a convergence type property. This is used in the proof of the results above but it is also of independent interest. In particular, if G acts co-compactly on X, then one obtains as a corollary that if the Tits diameter of ?X is bigger than 3π/2 then it is infinite and G contains a free subgroup of rank 2.     

Traces in Complex Hyperbolic Triangle Groups

We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant α of triangles in the complex hyperbolic plane. The main result of the paper is a formula, which expresses the trace of an element of the group as a Laurent polynomial in e^{i α} with coefficients independent of α and computable using a certain combinatorial winding number. We also give a recursion formula for these Laurent polynomials and generalise the trace formulas for the groups generated by complex μ-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.Research partially supported by NSF grant DMS-0072607 and by SFB 611 of the DFG.     

On Torsion-Free Minimal Abelian Groups

ABSTRACT

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. In this article we show that torsion-free groups which are complete in their ?-adic topology or are of p-rank not greater than 1, for all primes p, are minimal. A criterion is found for the minimality of all finite rank and for large classes of infinite rank completely decomposable groups. Separable minimal groups are also considered.     

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C² volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions. Partially funded by VIGRE grant DMS-9977371 Received: January 2005 Revision: August 2005 Accepted: September 2005     

On a theorem by Farb and Masur

Farb and Masur showed that an irreducible lattice in a semisimple Lie group of rank at least two always has finite image by a homomorphism into the outer automorphism group of a closed, orientable surface group. We point out that their theorem extends to the outer automorphism groups of a certain class of torsion-free, freely indecomposable word-hyperbolic groups.

     

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups \(\mathrm {PO}(p,q)\) by considering their action on the associated pseudo-Riemannian hyperbolic space \(\mathbb {H}^{p,q-1}\) in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.     

Torus Bundles over Locally Symmetric Varieties Associated to Cocycles of Discrete Groups

 We construct torus bundles over locally symmetric varieties associated to cocycles in the cohomology group , where Γ is a discrete subgroup of a semisimple Lie group and L is a lattice in a real vector space. We prove that such a torus bundle has a canonical complex structure and that the space of holomorphic forms of the highest degree on a fiber product of such bundles is isomorphic to the space of mixed automorphic forms of a certain type. (Received 4 September 1998)