Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
Completes results on complex braid group parabolic subgroups.
problem Proves properties of complex braid group parabolic subgroups.
method Uses Garside groupoid structure of B(G31) to extend results.
result Proves main theorems for B(G31) parabolic subgroups.
New proofs for growth series of Coxeter groups using complex structures.
problem Proving new formulae for growth series of Coxeter groups.
method Using the structure of Coxeter complexes, Davis complexes, or Tits non-complexes.
result Several classical formulae for growth series are proved in a new way.
Alternative construction of Rumin complex on Lie groups.
problem Constructing Rumin complex on homogeneous nilpotent Lie groups.
method Using ideas from parabolic geometry, an alternative construction to the classical one on Carnot groups.
result Explicit computations for the Engel group using the new approach.
The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.
problem Defining and characterizing parabolic subgroups in complex braid groups.
method Introducing and studying parabolic subgroups of generalized braid groups associated with complex reflection groups.
result Parabolic subgroups form a lattice in most cases, with specific properties and conjectures about hyperbolicity.
New complexity defined for groups, inspired by topological spaces.
problem Complexity of finitely presented groups.
method Inspired by Karoubi-Weibel work, introduces combinatorial complexity.
result New complexity (covering type) defined and properties considered.
New complex structures found on tangent bundles of Lie groups.
problem Finding integrable complex structures on tangent bundles of Lie groups.
method Inspired by Samelson's construction, a left-invariant integrable almost complex structure is defined on the tangent bundle of any compact Lie group.
result Tangent bundles of compact Lie groups admit left-invariant integrable almost complex structures.
Mapping class group subgroups yield quasi-isometric curve complex.
problem Understanding the curve complex through coset intersections.
method Proving quasi-isometry and combinatorial equivalence of curve complex and coset intersection complex.
result Automorphism group of coset intersection complex is the extended mapping class group.
Given a complex of groups over a finite simplicial complex in the sense of Haefliger, we give conditions under which it is possible to build an EZ-structure in the sense of Farrell-Lafont for its fundamental group out of such structures for its local groups. As an application, we prove a combination theorem that yields…
Complex Hyperbolic Triangle Groups of Type [m,m,0;n1,n2,2]math.GT The study determines discreteness of complex hyperbolic triangle groups.
problem Discreteness of complex hyperbolic triangle groups of specific type.
method Analysis of groups generated by complex reflections with given orders and distances.
result Determined intervals in parameter space for discrete and non-discrete groups.
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.
Study isotropy groups for complex orthogonal and skew-symmetric matrices.
problem Understanding isotropy subgroups of orthogonal similarity transformations.
method Analysis of group structure of nonsingular block matrices.
result Group structure of isotropy subgroups related to block Toeplitz matrices.
We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Co…
Division algorithm for surface group rings yields standard complexes and cohomological dimensions.
problem Understanding cohomological dimensions of surface group actions.
method Division algorithm for group rings of surface groups.
result Some 2-complexes with surface fundamental groups are standard.
The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simpl…
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.
The braid group of a complex reflection group is shown to be an index d subgroup.
problem Understanding the structure of braid groups associated with complex reflection groups.
method Presented a compatible presentation for the braid group of the orbifold quotient and a tagged triangulation of the disk.
result The braid group of the complex reflection group G(d,d,n) is an index d subgroup of the braid group of the orbifold quotient. Complex of cuts reveals full automorphism group for certain Stone spaces.
problem Understanding automorphism groups of Stone spaces.
method Introduced a complex of cuts and proved its significance.
result Full automorphism group identified for specific conditions.
We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…
Study quasi-isometry invariants of square complexes and their applications.
problem Classifying quasi-isometry types of 2D right-angled Artin groups and graph 2-braid groups.
method Define and analyze intersection complexes for universal covers of weakly special square complexes.
result Discover new quasi-isometric relationships between graph 2-braid groups and right-angled Artin groups.
New groups found with critical exponents close to but less than max.
problem Finding discrete isometry groups with critical exponents near maximum.
method Analyzing complex hyperbolic spaces to construct groups.
result Discrete isometry groups with critical exponents arbitrarily close to max but less.
The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.
problem Conditions for groups acting on CAT(0) cube complexes to have infinite girth.
method Analyzes lattices in automorphism groups of finite dimensional CAT(0) cube complexes.
result Groups either have infinite girth or are {locally finite}-by-{virtually abelian}.
In this paper we consider ultra-parallel complex hyperbolic triangle groups of type [m1,m2,0], i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness…
Study shows infinite manifold types for every group.
problem Finding manifold types for every finite group.
method Proved existence of infinitely many compact complex hyperbolic 2-manifolds.
result For every finite group, there are infinitely many isomorphism classes of compact complex hyperbolic 2-manifolds with automorphism group isomorphic to the group.
Study mapping class groups on CAT(0) cube complexes.
problem Asymptotically rigid mapping class groups of infinitely-punctured surfaces.
method Introduced CAT(0) cube complexes and determined their properties.
result Determined CAT(0) properties of cube complexes.
Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
The paper shows geometric realisation over specific groups and knots.
problem Geometric realisation of modules over aspherical groups and knots.
method Using extensions of scalars of relation modules and constructing specific 2-complexes.
result Exotic presentations of groups and stably free non-free modules over Baumslag-Solitar groups.
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
Study curvature of complex Finsler metrics on Lie groups.
problem Explicitly calculate and characterize curvature of complex Finsler metrics.
method Analyzes left-invariant complex Finsler metrics on Lie groups using their Lie algebra.
result Characterizes conditions for a Finsler metric to be Kähler or weakly Kähler.
CR embeddings in complex spaces for specific Lie groups.
problem Embedding specific Lie groups in complex spaces.
method Using integrable complex structures on subbundles of tangent bundles.
result CR embeddings possible as the edge of wedges in complex domains.
We investigate the notion of complexity for finitely presented groups and the related notion of complexity for three-dimensional manifolds. We give two-sided estimates on the complexity of all the Milnor groups (the finite groups with free action on the three-sphere), as well as for all finite Abelian groups. The ideas…
Proves a complex structure conjecture for a specific type of Lie groups.
problem Proving a conjecture about left-invariant complex structures on nilpotent Lie groups.
method Analyzes simply connected, nilpotent Lie groups of dimension 2n.
result Proves biholomorphism to C^n for the specified Lie groups.
Complex Hyperbolic Triangle Groups of Type [m,m,0;3,3,2]math.GT In this paper we study discreteness of complex hyperbolic triangle groups of type [m,m,0;3,3,2], i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders 3,3,2 in complex geodesics with pairwise distances m,m,0. For fixed m, the parameter space of such groups is…
Defines fundamental racks for braid spaces of complex reflection groups.
problem Understanding fundamental racks for braid spaces of complex reflection groups.
method Defines an augmented rack associated to the orbifold fundamental group.
result Yields representations of the orbifold fundamental group on the cohomology of the rack space.
New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid M(n,m) for (n,m)-torus knot groups and other braid groups. result New Garside structures for (n,m)-torus knot groups and related braid groups are constructed. These are lectures on discrete groups of isometries of complex hyperbolic spaces, aimed to discuss interactions between the function theory on complex hyperbolic manifolds and the theory of discrete groups.
Following an idea of Gonçalvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.
Compact complex manifolds with specific group actions are conformally flat.
problem Compact complex manifolds with invariant conformal holomorphic structures.
method Study of manifolds with transitive and essentially acting complex semi-simple Lie groups.
result If a complex semi-simple Lie group acts transitively and essentially, the manifold is conformally flat.
The paper studies complex manifolds using groups Gnk and Γnk.
problem Understanding complex manifolds through algebraic structures.
method Constructing a moduli space and studying its fundamental group mapped to Gnk and Γnk. result The fundamental group of the moduli space is naturally mapped to Gnk and Γnk, advancing the Gnk and Γnk approach. Finite stature proven for cube complexes with cyclonormal edges.
problem Understanding the structure of cube complexes with specific edge properties.
method Analyzing the fundamental groups of edge and vertex spaces, showing cyclonormality and virtual specialness.
result The fundamental group of a cube complex has finite stature with respect to vertex groups.
The paper introduces a Deligne complex for Artin monoids and studies its properties.
problem Understanding geometric structures associated with Artin monoids.
method Constructing a Deligne complex for Artin monoids and analyzing its properties.
result The Deligne complex for Artin monoids is contractible and can be embedded into the Deligne complex for the corresponding Artin group.
Study proves topological complexity and LS-category inequalities for specific groups and manifolds.
problem Proving inequalities for topological complexity and LS-category of specific groups and manifolds.
method Analyzing torsion free hyperbolic and nilpotent groups, lens spaces, using inequalities and counter-examples.
result Proves inequalities for topological complexity and LS-category of specific groups and manifolds.
We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) 2-complexes. We show, as an extension of an earlier work, that computing first homology of 2-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usua…
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
Picard modular groups are shown to be generated by complex reflections.
problem Understanding the structure of Picard modular groups using reflections.
method Using presentations from previous works to show generation by reflections.
result Picard modular groups mPU(2,1,Od) are generated by complex reflections. New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of ℓq,p cohomology results to all nilpotent Lie groups.