Characterizes unit ball using automorphism group size and boundary regularity.
problem Characterizing the unit ball in different projective spaces.
method Study projective automorphism groups and their relation to boundary regularity.
result Characterizes unit ball using automorphism group size and boundary regularity.
Compact automorphism group of a topological parallelism in real projective 3-space proved.
problem Proving compactness of automorphism group for topological parallelism.
method Direct proof without relying on earlier results.
result Automorphism group is compact.
Compactness proven for a group of transformations in 3D space.
problem Proving compactness of a group of transformations in 3D space.
method Conjecture and proof of identity component compactness.
result Identity component of the automorphism group is compact.
Finite groups can be automorphism groups of translation surfaces with poles.
problem Existence of finite automorphism groups on translation surfaces with poles.
method Analyzing translation surfaces with poles and extending results to branched projective structures.
result Finite groups can be automorphism groups of translation surfaces with poles.
Maximal automorphisms found for complex projective structures.
problem Maximizing automorphisms in complex projective structures.
method Analyzing Fuchsian uniformizations and Galois Belyi curves.
result Fuchsian uniformizations of Hurwitz surfaces achieve maximal automorphisms.
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
Describes automorphism group of Rauzy diagrams.
problem Understanding the structure of Rauzy diagrams.
method Using an example from Yoccoz's unpublished notes.
result Automorphism group described as a subgroup of symmetric group.
The curve complex of a 3-holed projective plane is shown to be quasi-isometric to a tree and its automorphisms are isomorphic to the mapping class group.
problem Characterizing the automorphisms of the curve complex of a 3-holed projective plane.
method Exhaustion of the curve complex by finite rigid sets, quasi-isometry to a tree, and isomorphism to the mapping class group.
result The group of simplicial automorphisms of the curve complex is isomorphic to the mapping class group.
The paper studies the non-discrete automorphisms of projective manifolds.
problem Analyzing non-discrete automorphisms of projective manifolds.
method Applying results from [13] and Benzekri's functor to study topological properties.
result Orbits of the connected component of automorphisms are immersed projective submanifolds.
New proof shows smallest non-cyclic automorphism group quotients are linear 2-groups.
problem Identifying the smallest non-cyclic quotients of free group automorphism groups.
method Elementary proof using standard projection and automorphisms.
result All minimal quotients are linear groups over the field of two elements.
Maps between automorphism groups are isomorphisms for free factor complexes.
problem Understanding the structure of automorphism groups of free factor complexes.
method Establishing isomorphisms between automorphism groups and automorphism groups of free factor complexes.
result Natural maps from mAut(Fn) to the automorphism group of the free-factor complex AFn are isomorphisms. New classification of 16D planes with specific automorphism groups.
problem Classifying 16-dimensional projective planes with certain automorphism groups.
method Detailed analysis and classification of 16-dimensional planes with specific automorphism properties.
result Classification of 16-dimensional planes with a group of dimension at least 35, excluding planes fixed by exactly one flag.
By using a notion of a geometric Dehn twist in ♯k(S2×S1), we prove that when projections of two Z-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the Z-splittings generate a free group of ra…
Improved bound on groups preserving Clifford parallelism in 4 dimensions.
problem Characterizing Clifford parallelism by automorphisms.
method Improving the bound on the dimension of groups preserving Clifford parallelism.
result Improved bound to 4 dimensions for groups preserving Clifford parallelism.
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 of hyperbolic directions in convex projective geometry.
problem Understanding properties of quasi-geodesics in convex projective geometry.
method Three perspectives: Hilbert metric, boundary projective geometry, and automorphisms.
result Relationship between different definitions of Morse and regular quasi-geodesics.
New parallelisms of 3D projective space found and classified.
problem Classifying parallelisms of 3D projective space with specific automorphism groups.
method Generalized existing constructions to include oriented parallelisms and showed that only Clifford parallelism remains for higher-dimensional automorphism groups.
result Only Clifford parallelism remains for topological parallelisms with automorphism groups of dimension 3 or larger.
Study projective representations of infinite-dimensional Hilbert-Lie groups.
problem Characterize and classify representations of Hilbert-Lie groups.
method Use covariance with respect to one-parameter groups of automorphisms and implement perturbation theory.
result Explicit determination of central extensions for projective representations.
Computes infinitesimal automorphisms for L-valued Higgs bundles, leading to DM stacks.
problem Computing infinitesimal automorphisms for Higgs bundles.
method Extending known results, using obstruction theory.
result Shows moduli stack of stable Higgs bundles is a DM stack.
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
We create a minimal triangulation of 5D real projective space.
problem Tackling the minimal triangulation of 5D real projective space.
method Constructing a 6-dimensional polytope with a highly symmetric automorphism group.
result Our construction uses the fewest number of vertices (24) for a triangulation of 5D real projective space.
The paper explores domains in flag manifolds with specific geometric properties.
problem Exploring bounded domains in flag manifolds with co-compactly acting automorphism groups.
method Analyzing projective automorphism groups and convexity conditions.
result No examples of bounded domains exist in many flag manifolds, but they do in some cases.
Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free …
Automorphisms of Hessenberg varieties are algebraic tori of dimension n-1.
problem Understanding the automorphisms of Hessenberg varieties.
method Analyzing the structure of automorphism groups of Hessenberg varieties.
result The reductive part of the identity component of the automorphism group of a connected Hessenberg variety is an algebraic torus of dimension n-1.
The paper studies automorphisms of RAAGs and RACGs, proving properties of their fixed subgroups.
problem Fixed subgroups of automorphisms of RAAGs and RACGs.
method Introducing coarse-median preserving automorphisms and proving properties of fixed subgroups.
result Fixed subgroups of RAAGs and RACGs are finitely generated, undistorted, and quasi-convex.
An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have C1 boundary, and have word hyperbolic divid…
Study automorphisms of free groups on hyperbolic plane actions.
problem Understanding automorphisms of two-generator free groups acting on hyperbolic spaces.
method Analyzing dynamical systems and polynomial automorphisms.
result Ergodicity and dense orbit properties of specific actions.
The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.
problem Exploring acylindrical actions on trees and their properties.
method Demonstrates criteria for preserving acylindrical hyperbolicity and analyzes the outer automorphism group of Baumsligar-Solitar groups.
result Proves acylindrical hyperbolicity of non-solvable Baumsligar-Solitar groups.
Study Picard groups of curves with symmetry, focusing on abelian groups and hyperelliptic curves.
problem Understanding the Picard groups of moduli spaces of curves with symmetry.
method Theory of symmetric mapping class groups, finitely generated Picard groups computation.
result Finitely generated Picard groups for moduli spaces of curves with abelian automorphisms.
The paper explores parallelisms and spreads in 3D projective space.
problem Characterizing parallelisms and spreads in PG(3,R).
method Introducing topological parallelisms, analyzing their properties and relationships with spreads.
result There are more oriented parallelisms than ordinary parallelisms, and the automorphism group is SO(3).
In this paper we construct and study a new 15-vertex triangulation X of the complex projective plane $\CP^2$. The automorphism group of X is isomorphic to S4×S3. We prove that the triangulation X is the minimal by the number of vertices triangulation of $\CP^2$ admitting a chess colouring of four-dimens…
New Riemann surfaces with unique end and infinite type are constructed.
problem Creating Riemann surfaces with infinite type and a unique end.
method Constructing Riemann surfaces on the Loch Ness monster with specific automorphism groups.
result Riemann surfaces with infinite type and a unique end can be algebraically described.
In this paper, we study a flag complex which is naturally associated to the Thurston theory of surface diffeomorphisms for compact connected orientable surfaces with boundary. The various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into this flag comp…
A connected Fano complex-contact manifold is isomorphic to the kaehlerian C-space of Boothby type with a natural complex-contact structure corresponding to a non-abelian simple complex Lie algebra if the contact line bundle is very ample. A. Beauville relaxed the provision to two assumptions that the contact line bundl…
We show that a projective manifold is stable if and only if the Mabuchi energy is proper on the space of algebraic metrics. We show that stability implies finite automorphism group.
The study finds conditions for quaternionic structures on symmetric spaces.
problem Conditions for quaternionic structures on symmetric spaces.
method Analysis of Lie group actions and representations.
result Symmetric spaces have invariant quaternionic structures under specific conditions.
Study on homeomorphisms preserving C1 curves on surfaces.
problem Characterizing homeomorphisms that preserve C1 curves. method Local conditions on induced map on projective tangent bundle.
result Characterization of Homeo1(S) for most closed surfaces. We solved a conjecture about braid group quotients being alternating groups.
problem Understanding the smallest non-trivial quotients of braid group commutator subgroups.
method Proved the conjecture about alternating groups as quotients, showed minimal quotient maps.
result Proved conjecture about braid group quotients being alternating groups.
Proves rigidity of homeomorphisms for lamination spaces.
problem Rigidity of homeomorphisms in lamination spaces.
method Analyzes homeomorphisms preserving geometric intersection.
result Two rigidity results for automorphism groups of lamination spaces.
Study invariants for Lagrangian equivalence problem on manifolds.
problem Understanding invariants for Lagrangian equivalence on manifolds.
method Geometric determination of second-order invariants and analysis of higher-order invariants.
result Determined geometrically second-order invariants and higher-order invariants for dimM≥2. We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…
The outer automorphism group Out(F_2g) of a free group on 2g generators naturally contains the mapping class group of a punctured surface as a subgroup. We define a subsurface projection of the sphere complex of the connected sum of n copies of S^1 x S^2 into the arc complex of the surface and use this to show that thi…
H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a conta…
Constructs Cartan geometries from automorphism behaviors.
problem Determining Cartan geometries from automorphism local behavior.
method Introduces a construction for Cartan geometries capturing automorphism local behavior.
result The sprawl uniquely characterizes Cartan geometries with equivalent local behavior.
Defines higher order spectra for complex manifolds with group actions and equations.
problem Understanding spectra of complex manifolds with group actions.
method Introduces higher order spectra, defines refinements, and provides Macdonald type equations.
result Macdonald type equations for higher order spectra of complex manifolds with group actions.
We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are in analogy with those constructed in \cite{CLM}, where the target group is the mapping class group of a surface. Toward this goal, we develop tools in the free group setting …
Defines currents relative to a free factor system and proves dynamics for fully irreducible automorphisms.
problem Understanding dynamics of fully irreducible automorphisms on projective relative currents.
method Defining currents relative to a free factor system and proving dynamics.
result Uniform north-south dynamics on a subspace of projective relative currents for fully irreducible automorphisms.
Study Higgs bundles on smooth projective varieties and their restrictions to curves.
problem Interplay between Higgs bundles on smooth projective varieties and their restrictions to curves.
method Investigate the restriction map of Higgs bundles and study branes in moduli spaces.
result Interconnectedness of Higgs bundles and branes on smooth projective varieties and their restrictions.