Two-Dimensional Artin Groups with CAT(0) Dimension Three

We exhibit 3-generator Artin groups which have finite two-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by semi-simple isometries on a two-dimensional CAT(0) complex. We prove that infinitely many of these groups are the fundamental groups of compact, non-positively curved 3-complexes. These examples show that the geometric dimension of a CAT(0) group may be strictly less than its CAT(0) dimension.     

Groups Acting on CAT(0) Square Complexes

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of Alexandrov) finite square complex and the vertex links of Ycontain no simple loop consisting of five edges, then any subgroup of π ₁Y either is virtually free Abelian or contains a free group of rank two. In addition we discuss when a group generated by two hyperbolic isometries contains a free group of rank two and when two points in the ideal boundary of a CAT(0) 2-complex at Tits distance π apart are the endpoints of a geodesic in the 2-complex.     

Rationally 4-periodic biquotients

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.     

Subgroup separability of the figure 8 knot group

It is shown that the fundamental groups of certain non-positively curved 2-complexes have the property that their quasiconvex subgroups are the intersections of finite index subgroups.As a consequence, every geometrically finite subgroup of the figure 8 knot group is the intersection of finite index subgroups. The same result holds for many other prime alternating link groups.     

Simplicity and superrigidity of twin building lattices

Kac–Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices. Dedicated to Jacques Tits with our admiration     

Some curious Kleinian groups and hyperbolic 5-manifolds

We present a new family of discrete subgroups ofSO (5, 1) isomorphic to lattices inSO (3, 1). In some of the examples the limit sets are wildly knotted 2-spheres. As an application we produce complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds.     

Pinching Surface Groups in Complex Hyperbolic Plane

We construct first examples of discrete geometrically finite subgroups of PU(21) which contain parabolic elements, and are isomorphic to surface groups of genus 2.     

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     

Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of . Received: October 7, 1999.     

Stallings’ folds for cube complexes

We describe a higher dimensional analogue of Stallings’ folding sequences for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups that act properly and cocompactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.     

Hermitian Morita Theory and Unitary Groups of Modules over Semisimple Artinian Rings

This article investigates isomorphisms between certain subgroups of the projective unitary groups of hermitian modules over semisimple Artinian rings with anti-structures. These subgroups contain the commutator subgroups of the projective unitary groups. Specifically, the article provides conditions under which these isomorphisms are induced by and the underlying rings are connected by hermitian Morita equivalences (HMEs). This article introduces the hyperbolic length of a module as well as the concept of generalized hyperbolic modules over simple Artinian rings, and over semisimple Artinian rings with anti-structures. The article shows that the stated isomorphisms are induced by HMEs if all the following conditions hold: (a) the hermitian forms are nonsingular and trace valued; (b) the modules in question are generalized hyperbolic; (c) the hyperbolic length equals three or is greater than or equal to five (hyperbolic length is greater than or equal to five in the semisimple case). Significantly, the condition of the hyperbolic length of a module greater than or equal to m is satisfied by a set of modules larger than or equal to those satisfying the condition of the Witt index of the module greater than or equal to m.     

A note on the characteristic classes of non-positively curved manifolds

In this expository note, we give a simple conceptual proof of the Hirzebruch proportionality principle for Pontrjagin numbers of non-positively curved locally symmetric spaces. We also establish (non)-vanishing results for Stiefel–Whitney and Pontrjagin numbers of (finite covers of) the Gromov–Thurston examples of compact negatively curved manifolds. A byproduct of our argument gives a constructive proof of a well-known result of Rohlin: every closed orientable 3-manifold bounds orientably. We mention some geometric corollaries: a lower bound for degrees of covers having tangential maps to the non-negatively curved duals and estimates for the complexity of some representations of certain uniform lattices.     

Embeddings of Kleinian Groups with Torsion

Rips conjectured that a non-clementary word hyperbolic group is cohopfian if and only if it is freely indecomposable. The results and examples in this paper show that cohopficity phenomenon in the case of word hyperbolic group with torsion is much more complicated than the conjecture. In particular, the cohopficity of such groups is not determined by the numbers of their ends and the cohopficity is not preserved by finite index subgroups. Our results and examples arise from Kleinian groups. Orbifold structures and orbifold maps are the new tools in our discussions. Received September, 16, 1999, Accepted September 14, 2000     

A cubical flat torus theorem and the bounded packing property

We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.     

Morse theory and conjugacy classes of finite subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F _n inside mapping class groups, Aut(), and Out() which have infinitely many conjugacy classes of finite-order elements.      

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

Anti-tori in Square Complex Groups

An anti-torus is a subgroup 〈a,b 〉 in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering space X such that a and b do not have any commuting nontrivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups Γ_p,l originally studied by Mozes [Israel J. Math. 90(1–3) (1995), 253–294]. It turns out that anti-tori in Γ_p,l directly correspond to non commuting pairs of Hamilton quaternions. Moreover, free anti-tori in Γ_p,l are related to free groups generated by two integer quaternions, and also to free subgroups of . As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not free.     

Subgroups of Word Hyperbolic Groups in Dimension 2

If G is a word hyperbolic group of cohomological dimension 2,then every subgroup of G of type FP₂ is also word hyperbolic.Isoperimetric inequalities are defined for groups of type FP₂and it is shown that the linear isoperimetric inequality inthis generalized context is equivalent to word hyperbolicity.A sufficient condition for hyperbolicity of a general graphis given along with an application to ‘relative hyperbolicity’.Finitely presented subgroups of Lyndon's small cancellationgroups of hyperbolic type are word hyperbolic. Finitely presentedsubgroups of hyperbolic 1-relator groups are hyperbolic. Finitelypresented subgroups of free Burnside groups are finite in thestable range.     

Generic measures for hyperbolic flows on non-compact spaces

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense G _δ -subset consisting of ergodic measures fully supported on the non-wandering set. We also treat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces.     

The boundary of a Busemann space

Let be a proper Busemann space. Then there is a well defined boundary, , for . Moreover, if is (Gromov) hyperbolic (resp. non-positively curved), then this boundary is homeomorphic to the hyperbolic (resp. non-positively curved) boundary.