Cross ratios and cubulations of hyperbolic groups

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Rank-one isometries of CAT(0) cube complexes and their centralisers

If G is a group acting geometrically on a CAT(0) cube complex X and if g ∈ G has infinite order, we show that exactly one of the following situations occurs: (i) g defines a rank-one isometry of X; (ii) the stable centraliser SCG (g) = {h ∈ G ∣ ∃ n ≥ 1, [h, gn ] = 1} of g is not virtually cyclic; (iii) Fix Y (gn ) is finite for every n ≥ 1 and the sequence (Fix Y (gn )) takes infinitely many values, where Y is a cubical component of the Roller boundary of X which contains an endpoint of an axis of g. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.

Elements of topiary art of reserved man-made parks of the second half of the XX century

Elements of topiary art were studied in eleven park-monuments of landscape art (PMLA) and five complex monuments of nature (CMN), created in the second half of the twentieth century. To the elements of topiary art belong: shaped plants, plant-borders, hedges, living walls, pylons, berso, bosquets, and parterres. In PMsLA “Bondaretsky” and “Vysokivsky” were found no elements of topiary art. In nine PMsLA and five CMsN of Ukrainian Polissya were present five elements of topiary art (plant-borders, hedges (low, medium, high), living walls, pylons and shaped plants (ball, pyramid, cone, cube, complex geometric figures). The most common are plant-borders formed from Buxus sempervirens L., trimmed hedges from Picea abies Karst. and Thuja occidentalis L., untrimmed hedges from Juniperus sabina L., Physocarpus opulifolius Maxim., and Sorbaria sorbifolia (L.) A.Br., pylons and shaped plants in the form of a sphere, cube, complex geometric shapes - from Thuja occidentalis L. 12 families, 24 genera, 24 species, and 3 cultivars represent the systematic structure of woody plants in the elements of topiary art. Deciduous species of woody plants, namely the family Rosaceae Juss, prevail. In the elements of topiary art of PMLA and CMN there are species that are protected by the IUCN Red List (58 %), belonging to two categories of rarity NT (4 %), LC (54 %). In terms of the height of woody plants in the elements of topiary art, trees and shrubs are represented in equal numbers. By height, among tree plants prevail trees of the first magnitude 26 % and medium bushes 29 %, whereas there are slightly fewer trees of the fourth magnitude (15%) and high bushes (22 %). The condition of woody plants of 24 species and three cultivars is good, except for woody plants where timely and proper care was not carried out (formation of longitudinal and transverse profiles and annual pruning). Some plants should be replaced because of loss of aesthetics due to age.

Automorphisms of Contact Graphs of CAT(0) Cube Complexes

We show that, under weak assumptions, the automorphism group of a $\textrm{CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen’s contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov’s theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim–Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated with Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.

Tits alternative for Artin groups of type FC

Given a group action on a finite-dimensional {\mathrm{CAT}(0)} cube complex, we give a simple criterion phrased purely in terms of cube stabilisers that ensures that the group satisfies the strong Tits alternative, provided that each vertex stabiliser satisfies the strong Tits alternative. We use it to prove that all Artin groups of type FC satisfy the strong Tits alternative.

Virtually Cocompactly Cubulated Artin–Tits Groups

We give a conjectural classification of virtually cocompactly cubulated Artin–Tits groups (i.e., having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin–Tits groups of spherical type, FC type, or two-dimensional type. A particular case is that for $n \geqslant 4$, the $n$-strand braid group is not virtually cocompactly cubulated.

Relative cubulations and groups with a 2-sphere boundary

We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of actions on cube complexes, analogous to the results of Markovic and Haïssinsky in the closed case.

Hyperspectral complex-domain image denoising: cube complex-domain BM3D (CCDBM3D) algorithm

We consider hyperspectral phase/amplitude imaging from hyperspectral complex-valued noisy observations. Block-matching and grouping of similar patches are main instruments of the proposed algorithms. The search neighborhood for similar patches spans both the spectral and 2D spatial dimensions. SVD analysis of 3D grouped patches is used for design of adaptive nonlocal bases. Simulation experiments demonstrate high efficiency of developed state-of-the-art algorithms.

Cube complexes and abelian subgroups of automorphism groups of RAAGs

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

The BNSR-invariants of the Houghton groups, concluded

We give a complete computation of the Bieri–Neumann–Strebel–Renz invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT (0) cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each 1 ≤ m ≤ n − 1, Hn admits a map onto ℤ whose kernel is of type Fm−1 but not Fm; moreover, no such kernel is ever of type Fn−1.