Flat Torus Theorem Research Articles

Median Sets of Isometries in CAT(0) Cube Complexes and Some Applications

In this paper, we associate with isometries of CAT(0) cube complexes specific subspaces, referred to as median sets, which play a similar role as minimizing sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralizers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.

A flat torus theorem for convex co‐compact actions of projective linear groups

In this paper, we consider discrete groups in PGL d ( R ) acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well-known flat torus theorem for CAT ( 0 ) spaces.

Solvable Subgroup Theorem for simplicial nonpositive curvature

Given a group [Formula: see text] with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of [Formula: see text] is finitely generated and virtually abelian of rank at most [Formula: see text]. In particular, this gives a new proof of the above theorem for systolic groups. The main tools used in the proof are the Product Decomposition Theorem and the Flat Torus Theorem.

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.

Periodic flats in CAT(0) cube complexes

One of the best known properties of a word-hyperbolic group, is that it cannot contain a subgroup isomorphic to Z Z. For a group G acting properly and cocompactly on a CAT(0) space X , it is known that G is word-hyperbolic if and only if X does not contain an isometrically embedded flat plane E . The “flat torus theorem” asserts that if G contains a subgroup H S Z Z, then there is a flat plane stabilized by H . One is led to the following problem:

On the coarse-geometric detection of subgroups

We generalize [20] to give sufficient conditions, primarily on coarse geometry, to ensure that a subset of a Cayley graph is a finite Hausdorff distance from a subgroup. Using this result, we prove a partial converse to the Flat Torus Theorem for CAT(0) groups. Also using this result, we give sufficient conditions for subgroups and splittings to be invariant under quasi-isometries.

Flats and the flat torus theorem in systolic spaces

We develop the theory of minimal surfaces in systolic complexes, which is a powerful tool in studying systolic complexes. We prove that flat minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for flat surfaces which implies minimality. We also prove that minimal surfaces are stable under small deformations of their boundaries. 20F65, 20F67; 53C21

Isometry groups of non-positively curved spaces: discrete subgroups

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

Abelian Subgroups of Garside Groups

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g −1 Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.

Diagram groups and directed 2-complexes: Homotopy and homology

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so-called rigidity of the R. Thompson's group F and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F ∞ ) each of which contains all countable diagram groups. We show how to compute minimal presentations and homology groups of a large class of diagram groups. We show that the Poincaré series of these groups are rational functions. We prove that all integer homology groups of all diagram groups are free Abelian.

On the classification of CAT(0) structures for the 4-string braid group

This paper is concerned with the class of so-called CAT(0) groups, namely, those groups that admit a geometric (i.e., properly discontinuous, co-compact, and isometric) action on some CAT(0) space. More precisely, we are interested in knowing to what extent it is feasible to classify the geometric CAT(0) actions of a given group (up to, say, equivariant homothety of the space). A notable example of such a classification is the flat torus theorem, which implies that the minimal geometric CAT(0) actions of the free abelian group Z (n ≥ 1) are precisely the free actions by translations of Euclidean space E. Typically, however, a given group will have uncountably many nonequivalent actions, making any chance of a complete classification rather slim. It is therefore reasonable to consider, for example, only those actions of a group on a space of minimal possible dimension—namely, the geometric dimension of the group. Thus, the geometric actions of the free group Fn, n ≥ 2, on 1-dimensional CAT(0) spaces (R-trees) are classified by the compact metric graphs of Euler characteristic 1− n. Even so, the variety of, say, the 2-dimensional CAT(0) structures for closed surface groups would seem to be vast, with many of these structures being nonplanar (see Section 5.1). By contrast, we are able to cite examples of CAT(0) groups of geometric dimension 2 that have no 2-dimensional CAT(0) structure [7; 3]. Other results in a similar vein are to be found in [2], [12], and [11]. Consider the n-string braid group Bn defined by the following presentation: Bn = 〈a1, a2 , . . . , an−1 | aiai+1ai = ai+1aiai+1 for i = 1, . . . , n− 2; aiaj = aj ai if |i − j | ≥ 2〉. The braid group Bn has geometric dimension n−1, and it is known to be CAT(0) (with a structure in dimension n−1) if n ≤ 5 [5; 4; 6]. The question of existence of a CAT(0) structure is open for n ≥ 6. We note that each braid group has infinite cyclic center (see [9]). The group B3 acts geometrically on the product T × R, where T is a trivalent tree, and up to an independant homothety of each factor of the product this is the unique minimal CAT(0) structure. This fact was used by Hanham in his thesis [12]. We give a proof here (see Theorem 4) that simply reduces the question to