A Geometric Rational Form for Artin Groups of FC Type

If A is an Artin group whose poset of finite type special subgroups is a flag complex, then A is said to be of FC type. Such groups act cocompactly on a CAT(0) cubical complex with finite type Artin groups as stabilizers. We use the geometry of this complex to obtain a rational normal form for the group.     

A topological characterisation of hyperbolic groups

We characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space. This is, in turn, equivalent to a convergence group for which every point of the space is a conical limit point.

     

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.     

Bounding reflection length in an affine Coxeter group

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections, and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely, we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on ℝ ⁿ is bounded above by 2n, and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.     

Geometric invariants of spaces with isolated flats

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats. The groups in question include word hyperbolic CAT(0) groups as well as geometrically finite Kleinian groups and numerous two-dimensional CAT(0) groups. For such a group we show that there is an intrinsic notion of a quasiconvex subgroup which is equivalent to the subgroup being undistorted. We also show that the visual boundary of the CAT(0) space is actually an invariant of the group. More generally, we show that each quasiconvex subgroup of such a group has a canonical limit set which is independent of the choice of overgroup.The main results in this article were established by Gromov and Short in the word hyperbolic setting and do not extend to arbitrary CAT(0) groups.     

Groups with infinitely many types of fixed subgroups

It is a theorem of Shor that ifG is a word-hyperbolic group, then up to isomrphism, only finitely many groups appear as fixed subgroups of automorphisms ofG. We give an example of a groupG acting freely and cocompactly on a CAT(0) square complex such that infinitely many non-isomorphic groups appear as fixed subgroups of automorphisms ofG. Consequently, Shor’s finiteness result does not hold if the negative curvature condition is relaxed to either biautomaticity or nonpositive curvature. D. T. Wise was supported by grants from FCAR and NSERC.     

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.     

Deformation of proper actions on reductive homogeneous spaces

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of standard compact quotients of G/H, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Γ keeps acting properly discontinuously and cocompactly on G/H. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n, 2)/U(n, 1) by Zariski-dense discrete subgroups of SO(2n, 2) acting properly discontinuously.     

Entropy of Hadamard spaces

We consider a geodesically complete and proper Hadamard metric measure space X endowed with a Borel measure. Assuming that there exists a certain non-amenable group of isometry of X which acts freely, properly discontinuously and cocompactly on X and preserves the measure we show that the topological entropy of the geodesic flow on the orbit space is positive.     

Local quasiconvexity of groups acting on small cancellation complexes

Given a group acting cellularly and cocompactly on a simply connected 2-complex, we provide a criterion establishing that all finitely generated subgroups have quasiconvex orbits. This work generalizes the “perimeter method”. As an application, we show that high-powered one-relator products A∗B/《rⁿ》 are coherent if A and B are coherent.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

Boundary dimension in negatively curved spaces

LetX be a negatively curved (Gromov hyperbolic) space. We construct a bound on dim X when a group of isometries acts cocompactly onX. We construct an example of a negatively curved space with infinite-dimensional boundary.     

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. Partially supported by MSRI, SFB256 and University of Maryland. Partially supported by MSRI, SFB256 and NSF DMS-9104134.     

Gauss sums over some matrix groups

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involve classical Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups, i.e., the groups of upper triangular matrices. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds of Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [R. Ferguson, C. Hoffman, F. Luca, A. Ostafe, I.E. Shparlinski, Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. 23 (2010) 501–513].     

Studies on Jacobi–Davidson,Rayleigh quotient iteration,inverse iteration generalized Davidson and Newton updates

We study Davidson‐type subspace eigensolvers. Correction equations of Jacobi–Davidson and several other schemes are reviewed. New correction equations are derived. A general correction equation is constructed, existing correction equations may be considered as special cases of this general equation. The main theme of this study is to identify the essential common ingredient that leads to the efficiency of a diverse form of Davidson‐type methods. We emphasize the importance of the approximate Rayleigh‐quotient‐iteration direction. Copyright © 2006 John Wiley & Sons, Ltd.     

The Auslander conjecture for NIL-affine crystallographic groups

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups in Aff(N)=NAut(N) acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions 1n5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.     

Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

Abstract

We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.     

ℓ-homology of right-angled Coxeter groups based on barycentric subdivisions

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Σ_L on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a triangulation of , then Σ_L is a contractible n-manifold. We establish vanishing (in a certain range) of the reduced ℓ²-homology of Σ_L in the case where L is the barycentric subdivision of a cellulation of a manifold. In particular, we prove the Singer Conjecture (on the vanishing of the reduced ℓ²-homology except in the middle dimension) in the case of Σ_L where L is the barycentric subdivision of a cellulation of , n=6,8.     

Schwartz groups and convergence of characters

It is proved that a locally quasi-convex group is a Schwartz group if and only if every continuously convergent filter on its dual group converges locally uniformly. We also show that for metrizable separable groups a similar result remains true when filters are replaced by sequences. As an ingredient in the proofs of these results, we obtain a Schauder-type theorem on compact homomorphisms acting between the natural group analogues of normed spaces.     

Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups which can act properly discontinuously, isometrically, and cocompactly on hyperbolic -space in terms of the combinatorics of the action of on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the -sphere.