Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.
problem Existence of non-residually finite hyperbolic groups
method Direct implication
result Existence of non-residually finite rigid hyperbolic groups
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.
Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Finite rigid sets found in surface curve complexes.
problem Finding rigid sets in surface curve complexes.
method Incidence-preserving maps to find rigid subcomplexes.
result Finite rigid subcomplexes identified in surface curve complexes.
Finite rigid sets found in sphere complexes for some but not all cases.
problem Characterizing finite rigid sets in sphere complexes.
method Analyzing locally injective maps and automorphisms.
result Finite rigid sets exist for n≥3 but not for n=2. For an orientable surface S of finite topological type with genus g≥3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of S. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a …
In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…
New examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
New insights into profinite rigidity of Kleinian groups and their subgroups.
problem Characterizing profinite completions of Kleinian groups and their subgroups.
method Analyzing profinite completions and using finite index subgroups to distinguish completions.
result Profinite completions of certain subgroups of finite index in Kleinian groups are not isomorphic.
Let S be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex C(S) by a sequence of finite rigid sets.
Study on finiteness properties of handlebody mapping class groups.
problem Understanding finiteness properties of asymptotically rigid handlebody groups.
method Introduced asymptotically rigid mapping class groups and determined their finiteness properties based on the space of ends of handlebodies.
result Homology of these groups coincides with stable homology of handlebody groups in some cases.
3-manifold groups are uniquely identifiable via their profinite completions.
problem Identifying uniquely 3-manifold groups among finitely generated groups.
method Proving that the inclusion of profinite completions is not an isomorphism for proper subgroups.
result All finitely generated 3-manifold groups are Grothendieck rigid.
In this article we use the "escape from subvarieties lemma" introduced by Eskin--Mozes--Oh to prove finite step rigidity results for the Jordan-Lyapunov projection spectra of Hitchin representations and the Margulis-Smilga invariant spectra of some special Margulis-Smilga spacetimes. In the process, we also prove a sim…
This paper exhausts curve complexes on non-orientable surfaces.
problem Proving exhaustion of curve complexes on non-orientable surfaces.
method Proving exhaustion via rigid expansions and graph endomorphisms.
result Any graph endomorphism of curve complexes whose restriction to a finite rigid set is injective is induced by a homeomorphism.
In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature −b2≤K≤−1 and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature …
We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping cl…
Proof that specific groups are quasi-isometrically rigid.
problem Quasi-isometric rigidity of specific groups.
method Proof of quasi-isometric rigidity for Kleinian and three-manifold groups.
result Finitely generated Kleinian and three-manifold groups are quasi-isometrically rigid.
Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group PSL(2,Z[ω]) with ω2+ω+1=0 is rigid in this sense. Other examples include th…
For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface wit…
Study shows certain infinite-ended groups are not quasi-isometrically rigid.
problem Understanding when infinite-ended groups are quasi-isometrically rigid.
method Combining results on subgroups and hyperbolic groups, adapting Whyte's argument.
result Proves certain infinite-ended groups are not quasi-isometrically rigid.
A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected non-orientable surfaces of genus g with n holes …
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.
problem Understanding actions of finite groups on aspherical manifolds.
method Analyzing the outer automorphism group and homeomorphism group of the fundamental group.
result Proves the homeomorphism group is Jordan and bounds the discrete degree of symmetry.
We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.
Study proves rigidity of critical points in hydrophobic capillary systems.
problem Rigidity of critical points in hydrophobic capillary systems.
method Proves rigidity among sets of finite perimeter in the half space, extending to full hydrophobic regime.
result Rigidity of critical points proven in hydrophobic capillary systems.
Constructs surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
problem Creating surfaces that can be tiled by a finite set of rigid motion congruence classes of tiles.
method Constructs examples with various topologies and describes all monotilings by finite edge prototiles.
result Describes all monotilings by finite edge prototiles with three or less edges.
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
Aramayona and Leininger have provided a "finite rigid subset" X(Σ) of the curve complex C(Σ) of a surface Σ=Σgn, characterized by the fact that any simplicial injection X(Σ)→C(Σ) is induced by a unique element of the mapping class group Mod(Σ). In this…
Theory of symmetric rigidity in hyperbolic geometry.
problem Symmetric rigidity in hyperbolic frameworks.
method Gain graphs and orbit rigidity matrix, matroidal sparsity conditions.
result Characterization of infinitesimal rigidity for Gamma-symmetric frameworks.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
This study exhausts curve graphs of low-genus surfaces.
problem Exhausting curve graphs of low-genus surfaces.
method Constructing finite subgraphs and using rigid expansions.
result Graph morphisms and endomorphisms are automorphisms and induced by homeomorphisms.
Profinite rigidity of certain 3-manifolds detected through Dehn fillings.
problem Detecting profinite rigidity in 3-manifolds.
method Observation of profinite isomorphisms between cusped hyperbolic 3-manifolds and their Dehn fillings.
result Some cusped hyperbolic 3-manifolds are profinitely rigid.
Study shows only grim reaper cylinder for certain self-translating surfaces.
problem Characterizing self-translating surfaces in 3D space.
method Used parabolicity in a weighted setting and universally L-superharmonic functions.
result Characterized the grim reaper cylinder as the only finite entropy self-translating 2-surface in R^3 of width π and bounded from below.
Survey on quasi-isometries of group pairs and their invariants.
problem Understanding quasi-isometries of group pairs and their invariants.
method Exploration of quasi-isometry and qi-characteristic collections of subgroups.
result New insights into phenomena observed in quasi-isometric rigidity.
We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X into C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) …
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤π, possibly with boundary consisting of totally geodesic hyperbo…
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorit…
Researchers solve 3D cube complex boundary rigidity problem.
problem Determining the combinatorial type of a 3D CAT(0) cube complex from boundary distances.
method Discrete version of boundary rigidity problem, focusing on CAT(0) cube complexes.
result The combinatorial type of a finite CAT(0) cube complex can be reconstructed from its boundary distances.
The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
Let N be a compact, connected, nonorientable surface of genus g with n boundary components. Let C(N) be the curve complex of N. We prove that if (g,n)=(3,0) or g+n≥5, then there is an exhaustion of C(N) by a sequence of finite rigid sets. This improves the author's result on…
We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any n≥4, we construct a finite subgraph Xn of the pants graph P(S0,n) of the n-punctured sphere S0,n with the following property. Any simplicial embedding of Xn into any pants graph P(S0,m) of a punctured …
5D shrinking solitons with bounded curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons.
method Proving rigidity for solitons with bounded curvature.
result 5D shrinking gradient Ricci solitons with bounded curvature are rigid.
Study on profinite rigidity of direct products of free and surface groups.
problem Profinite rigidity of direct products in residually free groups.
method Rank gradients of pro-p groups, virtual homology of limit groups. result Direct products of free and surface groups are profinitely rigid.
Groups with similar pro-p completions have two-generated subgroups that are free.
problem Understanding the profinite rigidity of groups with specific pro-p completions. method Utilizing new results on residually-(torsion-free nilpotent) groups and their subgroups, and applying these to prove profinite rigidity.
result Groups with the same pro-p completion have two-generated subgroups that are free.