Rank Rigidity for Cat(0) Cube Complexes

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rankone isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.     

Actions of Picture Groups on CAT(0) Cubical Complexes

A class of groups, called picture groups, is defined. Richard Thompsons groups F, T, and V are all picture groups. Each picture group is shown to act properly and isometrically on a CAT(0) cubical complex. In particular, all picture groups are a-T-menable.Mathematics Subject Classifications (2000). 20F65, 43A15.     

Limit Groups are Cat(0)

We prove that every limit group acts geometrically on a CAT(0)space with the isolated flats property.     

The Tits Alternative for Groups Defined by Periodic Paired Relations

The class of groups defined by periodic paired relations, as introduced by Vinberg, includes the generalized triangle groups, the generalized tetrahedron groups, and the generalized Coxeter groups. We observe that any group defined by periodic paired relations Γ can be realized as a so-called “Pride group”. Using results of Howie and Kopteva we give necessary and sufficient conditions for this Pride group to be non-spherical. Under such conditions, we show that Γ satisfies the Tits alternative.

Communicated by A. Olshanskiy     

Complex Surfaces with Cat(0) Metrics

We study complex surfaces with locally CAT(0) polyhedral K?hler metrics and construct such metrics on \mathbbCP²{\mathbb{C}P^{2}} with various orbifold structures. In particular, in relation to questions of Gromov and Davis–Moussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of \mathbbCP²{\mathbb{C}P^{2}} of sufficiently high degree their desingularizations are of type K(π, 1).     

Cat(0) Groups with Non-Locally Connected Boundary

The main theorem shows that whenever certain amalgamated productsact geometrically on a CAT(0) space, the space has non-locallyconnected boundary. One can now easily construct a wide varietyof examples of one-ended CAT(0) groups with non-locally connectedboundary. Applications of this theorem to right-angled Coxeterand Artin groups are considered. In particular, it is shownthat the natural CAT(0) space on which a right-angled Artingroup acts has locally connected boundary if and only if thegroup is Zⁿ for some n.     

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.     

The Tits alternative for non-spherical Pride groups

Pride groups, or ‘groups given by presentations in whicheach defining relator involves at most two types of generators’,include Coxeter groups, Artin groups, triangles of groups, andVinberg's groups defined by periodic paired relations. We showthat every non-spherical Pride group that is not a triangleof groups satisfies the Tits alternative.     

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on S ¹ . In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group exists if and only if the quotient group does not contain a free subgroup with two generators (here is some specific subgroup of G defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups of homeomorphisms of the line.     

(Min,max)-equivalence of posets and nonnegative Tits forms

We study the relationship between the (min, max)-equivalence of posets and properties of their quadratic Tits form related to nonnegative definiteness. In particular, we prove that the Tits form of a poset S is nonnegative definite if and only if the Tits form of any poset (min, max)-equivalent to S is weakly nonnegative. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1157–1167, September, 2008.     

超球上可微(0,q)式(-e)-问题的解

证明了:对任一在Cn中超球Bn上可微的(0,q)式g(z)若适合(-e)g=0,而且g在Bn内有紧致的支集,则当n≥q+1时v(w)=(-e)*∫Bng(z)∧*z—N(z,w)是(-e)方程(-e)v=g之解,它在边界Bn上为零.     

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E ₆.     

ℵ0-quasibarrelledness onc 0 (E)

Archiv der Mathematik -     

The (2, 2, 0) group inverse problem

We characterize the existence of the group inverse of a two by two matrix with zero (2, 2) entry, over a ring by means of the existence of the inverse of a suitable function of the other three entries. Some special cases are derived.