Paper proves isotopic 2-knots can be transformed by cubulated moves.
problem Determining isotopy of 2-knots in 4D.
method Finite sequence of cubulated moves analogous to Reidemeister and Roseman moves.
result Two cubical links in 4D are isotopic if and only if one can pass to the other by cubulated moves.
The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism gro…
In this paper, we prove than given two cubic knots K1, K2 in R3, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by…
Study PL topology theorems for cubical complexes, solving Habegger and Funar's conjecture.
problem Characterize PL homeomorphic cubulations equivalence by Pachner moves.
method Show equivalence to the existence of cobordisms between generic immersions of hypersurfaces.
result Solve Habegger and Funar's conjecture about PL homeomorphic cubulations equivalence.
Random quotients of hyperbolic cubulated groups remain cubulated.
problem Understanding properties of random quotients of hyperbolic cubulated groups.
method Cubical small-cancellation theory, exponential growth of conjugacy classes, and hyperplane stabilizers' growth.
result Low-density random quotients of cubulated hyperbolic groups are cubulated and hyperbolic.
The paper describes how to deform cubulations of hyperbolic groups.
problem The challenge is to understand and construct deformations of cubulations in hyperbolic groups.
method The approach involves bending hyperplanes to deform cubulations, inspired by various examples in hyperbolic geometry.
result Every cocompactly cubulated hyperbolic group admits infinitely many bald cubulations, except for certain specific groups.
Clarifies boundary criterion for non-one-ended subgroups in cubulation theory.
problem Boundary criterion for relative cubulation in non-one-ended subgroups.
method Showed that if boundary criterion is satisfied for a relatively hyperbolic group, the group admits a relatively geometric action on a CAT(0) cube complex.
result The refinement of the boundary criterion is useful for constructing new relative cubulations.
We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical C′(1/6)--small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first ins…
New groups act on cube complexes without compact cubulation.
problem Triangle-free Artin groups without compact cubulation.
method Proved proper actions on CAT(0) cube complexes.
result First examples of non-cocompactly cubulated groups.
Accessible proof of hyperbolic cubulated groups being virtually special.
problem Proving hyperbolic cubulated groups are virtually special.
method Retained Agol's ideas but made the proof more transparent and detailed.
result Accessible proof of the main theorem.
The paper proves drilled bundles over graphs are virtually special cubulable.
problem Proving drilled bundles over graphs are virtually special cubulable.
method Starting with a Gromov-hyperbolic surface bundle, drilling out essential curves, and using relative hyperbolicity and Wise's theorem.
result Proves drilled bundles over graphs are virtually special cubulable.
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
Study of cubulations in hyperbolic groups and their invariant cross ratios.
problem Understanding geometric structures in hyperbolic groups.
method Analysis of cubulations and cross ratios in Gromov hyperbolic groups.
result Essential cubulations of hyperbolic groups are length-spectrum rigid.
New findings on hyperbolic groups and their boundaries.
problem Understanding the structure of cubulated hyperbolic groups with specific boundary conditions.
method Utilizing ideas from Markovic's work on Cannon's conjecture, focusing on quasi-convex subgroups and limit sets.
result Cubulated hyperbolic groups with certain boundary conditions are virtually fundamental groups of specific manifolds.
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≥4, the n-st…
The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a ver…
The paper proves conditions for cubulating surface-by-free groups.
problem Proving conditions for cubulating surface-by-free groups.
method Combination theorem for virtually special hyperbolic groups, tracks theory, quasiconvex hierarchy theorem, distance estimates, and model geometry.
result Main result: sufficient conditions for cubulability of G. New hyperbolic 3-pseudomanifolds with unique properties.
problem Understanding cubulable groups in hyperbolic 3-manifolds.
method Constructing compact hyperbolic 3-manifolds with specific boundary conditions.
result Found groups that are word hyperbolic but not cubulable.
We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly, we prove that a closed aspherical Kähler manifold with a cu…
We find explicit subdivision rules for all special cubulated groups. A subdivision rule for a group produces a sequence of tilings on a sphere which encode all quasi-isometric information for a group. We show how these tilings detect properties such as growth, ends, divergence, etc. We include figures of several worked…
Study of group actions on CAT(0) cube complexes, focusing on marked length spectra.
problem Comparing marked length spectra of group actions on CAT(0) cube complexes.
method Use of finite-state automata and thermodynamic formalism for suspension flows over subshifts of finite type.
result Prove that the Manhattan curve is analytic and convex, and a straight line if and only if marked length spectra are homothetic.
The study shows that surface groups are the only non-free infinite index subgroups of certain hyperbolic groups.
problem Identifying the only non-free infinite index subgroups of specific hyperbolic and one-relator groups.
method Careful analysis of free and cyclic splittings of cubulated groups.
result Proves that surface groups are the only non-free infinite index subgroups of certain hyperbolic and one-relator groups.
Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
We describe a correspondence between spaces with walls and CAT(0) cube complexes.
We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds wit…
Minimal area of spun trefoil knot is found in 4D cubical space.
problem Finding the minimum area for a spun trefoil knot in 4D cubical space.
method Defined minimal area of cubical 2-knots, used isotopy to find minimum area for spun trefoil.
result Spun trefoil knot requires a specific minimal area in 4D cubical space.
Stable cubulations and bicombings in mapping class groups and Teichmüller spaces.
problem Understanding geometric structures in mapping class groups and Teichmüller spaces.
method Proving stably approximated by CAT(0) cube complexes, applying to broader colorable hierarchically hyperbolic spaces and groups.
result Stable cubulations and bicombings in mapping class groups and Teichmüller spaces, with stable coarse barycenters.
A tubular group is a group that acts on a tree with Z2 vertex stabilizers and Z edge stabilizers. This paper develops further a criterion of Wise and determines when a tubular group acts freely on a finite dimensional CAT(0) cube complex. As a consequence we offer a unified explanation of the fai…
The paper shows how sublinearly Morse boundaries can be understood through combinatorial methods.
problem Understanding sublinearly Morse boundaries in cubulated groups and CAT(0) cube complexes.
method Combining geometric and combinatorial approaches to analyze sublinearly Morse boundaries.
result Sublinearly Morse boundaries can be described combinatorially and continuously related to Gromov and Roller boundaries.
We bound the size of d-dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.
Effective rank rigidity proved for cubulated groups with factor systems.
problem Rank rigidity in cubulated groups with factor systems.
method Exhibiting special pairs of hyperplanes and curtains for skewering.
result Effective form of rank rigidity proved for cubulated groups.
Let M be a graph manifold. We show that π_1M is the fundamental group of a compact nonpositively curved cube complex if and only if M is chargeless. We also prove that in that case π_1M is virtually compact special.
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
New group acts on complex but not in lower dimensions.
problem Understanding group actions on cube complexes.
method Constructed a specific group acting on a locally finite CAT(0) cube complex.
result Group does not act properly on a finite dimensional CAT(0) cube complex.
New hyperbolic manifolds with diverse features created.
problem Creating new types of hyperbolic manifolds.
method Using cubulations and hyperbolization procedures.
result First examples of hyperbolic manifolds with non-trivial Stiefel-Whitney and Pontryagin classes.
Proves small cancellation free products have geometric actions on CAT(0) cube complexes.
problem Proving small cancellation free products have geometric actions on CAT(0) cube complexes.
method Using a blown-up complex of groups and a boundary separation criterion, proving wall stabilizers form a rich family of subgroups.
result Proves $C'( rac16)$--small cancellation free products of residually finite groups are residually finite.
New method approximates hyperbolic lattices using cube complexes.
problem Metric approximation of hyperbolic lattices by cubulations.
method Study of co-geodesic currents and their intersection number.
result Isometric actions of hyperbolic lattices can be approximated by geometric actions on CAT(0) cube complexes.
We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions us…
The study establishes conditions for groups acting on polygonal complexes to contain virtually free subgroups.
problem Conditions for groups acting on polygonal complexes to contain virtually free subgroups.
method Analysis of links of polygonal complexes and conditions on their structure.
result Groups acting on polygonal complexes with certain link conditions contain virtually free subgroups.
We say that a topological n-manifold N is a cubical n-manifold if it is contained in the n-skeleton of the canonical cubulation C of Rn+k (k≥1). In this paper, we prove that any closed, oriented cubical 2-manifold has a transverse field of 2-planes in the sense of Whitehead an…
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Ka…
New group constructed from cube complex properties.
problem Creating a new group from cube complex properties.
method Cubical Rips construction for finitely presented groups.
result New group surjects onto a given group with specific properties.
Groups acting on CAT(0) cube complexes have hyperfinite boundary actions.
problem Hyperfiniteness of boundary actions for special groups.
method Proving hyperfiniteness for groups acting on CAT(0) cube complexes.
result Boundary actions of virtually special groups are hyperfinite.
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken …
Cube complexes allow hyperbolic groups to have Anosov representations.
problem Understanding representations of hyperbolic groups in higher dimensions.
method Analyzing representations of hyperbolic groups acting on CAT(0) cube complexes.
result Generic representations of certain groups are Anosov.
New theorem bounds group quotient size to subgroups index.
problem Understanding subgroup structure in hyperbolic groups.
method Proved quotient size bounds on Eilenberg-MacLane spaces.
result One-ended hyperbolic groups cannot have isomorphic finite-index subgroups.
New groups with special properties found.
problem Finding new groups with specific geometric properties.
method Proved actions on CAT(0) cubical complexes under certain conditions.
result Many groups admit cocompact actions on CAT(0) cubical complexes.
Our main result is that for densities <103 a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure o…