The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.
problem Establishing geometrical finiteness for automorphism groups of K3 surfaces and related varieties.
method Using cone conjecture, the paper establishes geometrical finiteness for the natural isometric actions of automorphism groups on hyperbolic spaces.
result Automorphism groups of K3 surfaces and related varieties are non-positively curved and relatively hyperbolic.
The paper shows how surjective homomorphisms between surface braid groups factor and computes their automorphism groups.
problem Characterizing surjective homomorphisms and automorphisms of surface braid groups.
method Analyzing the structure of surface braid groups and their homomorphisms.
result Surjective homomorphisms factor through forgetful maps and automorphisms are geometric.
Every normal subgroup of Cantor tree's mapping class group is geometric.
problem Characterizing normal subgroups of mapping class groups.
method Generalized curve graph study and adaptation of Brendle-Margalit strategy.
result All normal subgroups of Cantor tree's mapping class group are geometric.
Random free group outer automorphisms are geometric and have nongeometric attracting trees.
problem Understanding the structure of random outer automorphisms of free groups.
method Analyzing the Whitehead graph and ideal Whitehead graph of random outer automorphisms.
result The attracting tree of a random outer automorphism is a nongeometric R-tree with all branch points trivalent.
This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemanni…
Proves conjecture for hyperbolic-by-cyclic groups using geometric methods.
problem Proving Farrell-Jones Conjecture for specific group structures.
method Geometric methods and structure theory of mapping tori.
result Proved Farrell-Jones Conjecture for mapping tori of automorphisms of hyperbolic-by-cyclic groups.
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.
Study on symplectic 4-manifolds and their automorphism groups.
problem Understanding when the subgroup of orientation-preserving diffeomorphisms has infinite index in the automorphism group.
method Analyzing the intersection form and its automorphisms for symplectic 4-manifolds.
result Identified conditions under which the subgroup of orientation-preserving diffeomorphisms has infinite index in the automorphism group.
Survey various symmetry notions for toric varieties.
problem Understanding different types of symmetries in toric varieties.
method Exploring algebraic, complex, representation, combinatorial, convex, and geometric stability perspectives.
result Establishes relationships between different symmetry notions.
The paper studies automorphisms of Weyl manifolds and constructs modified contact Weyl diffeomorphisms.
problem Analyzing the automorphisms of Weyl manifolds associated with symplectic structures.
method Investigates the automorphisms of Weyl manifolds corresponding to Poincaré-Cartan classes and constructs modified contact Weyl diffeomorphisms.
result Construction of modified contact Weyl diffeomorphisms and analysis of automorphisms of Weyl manifolds.
Complex of separating meridians can be extended to all meridians in genus ≥6 handlebodies.
problem Extending automorphisms of separating meridians to all meridians in handlebodies.
method Showing every automorphism of the complex of separating meridians can be extended to an automorphism on the complex of all meridians.
result Automorphisms of separating meridians can be extended to all meridians in handlebodies of genus ≥6.
New flat Minkowski planes created from convex functions.
problem Creating new geometric structures from convex functions.
method Constructing flat Minkowski planes using convex functions.
result Automorphism groups of these planes are at least 3-dimensional.
A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.
For a right-angled Artin group AΓ, the untwisted outer automorphism group U(AΓ) is the subgroup of Out(AΓ) generated by all of the Laurence-Servatius generators except twists (where a {\em twist} is an automorphisms of the form v↦vw with vw=wv). We define a space ΣΓ on which U(AΓ) acts properl…
The paper shows symmetries of a geometric space for Coxeter groups.
problem Understanding symmetries in the Outer space of a Coxeter group.
method Analyzing the geometric rigidity of the universal Coxeter group of rank n.
result For n ≥ 4, the symmetries of the spine of the outer space are only the outer automorphisms.
Groups of birational transformations with specific properties are conjugate to groups of pseudo-automorphisms.
problem Classifying groups of birational transformations with fixed point properties.
method Importing ideas from geometric group theory, proving conjugacy to groups of pseudo-automorphisms.
result Groups of birational transformations with certain properties are conjugate to groups of pseudo-automorphisms.
Many normal subgroups of mapping class groups are geometric.
problem Characterizing normal subgroups of mapping class groups of surfaces with punctures.
method Proving automorphism and commensurator groups of certain subgroups are isomorphic to the mapping class group, using simplicial complexes.
result Many normal subgroups of mapping class groups are geometric.
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
Study on automorphism groups of Coxeter groups, proving they are not CAT(0).
problem Determining the curvature of automorphism groups of Coxeter groups.
method Combinatorial and geometric analysis of automorphism groups of universal right-angled Coxeter groups.
result Proved that the natural model space for outer automorphism groups is not CAT(0).
Study on foliation automorphisms, finding non-Lie groups and ILH Lie groups.
problem Understanding the structure of diffeomorphism groups of foliations.
method Investigation of diffeomorphism groups of foliations, proving properties of automorphism groups.
result Found examples of foliations with non-Lie automorphism groups and proved properties of ILH Lie groups for certain foliations.
Study connects group invariants through outer automorphisms and polynomial relations.
problem Understanding polynomial invariants of free-by-cyclic groups.
method Introducing orientable fully irreducible outer automorphisms to relate McMullen polynomial and Alexander polynomial.
result Characterization of when homological stretch factor equals geometric stretch factor.
Characterizes groups arising as fixed subgroups of RAAG automorphisms.
problem Identifying groups that can be fixed by finite-order automorphisms of RAAGs.
method Geometric characterisation using divisible cube complexes.
result Surface groups and commutator subgroups of RAAGs are fixed subgroups.
We finish proving that an irreducible automorphism f of a handlebody is efficient if, and only if, a certain standard pair of dual f--invariant laminations have the geometric tightness property. In a previous paper it was proved that this tightness property implies efficiency. We now prove the converse.
Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.
problem Analyzing actions of Dehn twists in geometric group theory.
method Application of Dynnikov coordinates to describe orbits and dynamics of Dehn twists in a thrice-punctured disc.
result The action of Dehn twists has a geometric meaning as a piecewise linear Z2-automorphism. We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as…
Study on a metric for disk automorphisms with maximal modulus.
problem Characterizing the metric on disk automorphisms.
method Explicit formula for the metric induced by maximal modulus.
result Characterized almost regular Finsler structure.
New technique shows any group can be a map's automorphism.
problem Finding maps with any given automorphism group.
method Universal technique for showing any finite automorphism group is possible.
result Any finite automorphism group can be realized by many non-isomorphic maps.
We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…
The paper studies groupoid structures from different viewpoints.
problem Understanding groupoid morphisms and their structures.
method Constructing two groupoids from morphisms of groupoids, one from a categorical viewpoint and the other from a geometric viewpoint. Showing equivalence of the two kinds of groupoids of morphisms.
result Equivalence of two kinds of groupoids of morphisms for each pair of groupoids.
The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of…
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…
Study automorphism groups of right-angled Artin groups, proving they are type VF.
problem Characterize the outer automorphism groups of right-angled Artin groups.
method Construct subnormal series, study restriction homomorphisms, refine previous work.
result Prove Out(AΓ) is type VF with finite index subgroup having finite classifying space. Geometric models for Lie--Hamilton systems on \(\mathbb{R}^2\) are described.
problem Analyzing Lie--Hamilton systems on \(\mathbb{R}^2\).
method Two geometric models: 1) restriction to symplectic leaves, 2) projection onto quotient space.
result Natural framework for Lie--Hamilton systems on \(\mathbb{R}^2\).
Study quantization of Kähler-Einstein metrics using balanced metrics.
problem Approximating Kähler-Einstein metrics by balanced metrics.
method Use canonical Bergman metrics and introduce algebro-geometric obstructions.
result Existence and weak convergence of balanced metrics for CKE manifolds.
The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.
problem Classifying 3D spherical Sasakian manifolds with specific properties.
method Establishing correspondence between different sets of parameters and geometrically describing the moduli space.
result Determination of Sasakian automorphism groups and detection of homogeneous Sasakian manifolds.
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.
For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometr…
A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is …
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of areas of complex geometric analysis. Relations with other parts of mathematics are …
We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.
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}.
The study classifies subgroups of outer automorphisms of free products.
problem Classifying subgroups of outer automorphisms of free products.
method Geometric tool: boundaries of relative factor graphs and equivalence classes of arational trees.
result Every finitely generated subgroup either contains a relatively fully irreducible automorphism or virtually preserves a conjugacy class.
By analyzing how the Borel regulator classes vanish on various groups related to GL(n,Z), we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the IA-automorphism groups and we show that our classes coincide with…
The equivariant holomorphic torsion of a compact locally symmetric manifold and an automorphism is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.
Geometric model for a specific group in Artin groups.
problem Understanding the structure of outer automorphism groups in Artin groups.
method Proving proper cocompact action on a subcomplex of the spine of outer space.
result U(A; G, Ht) acts properly cocompactly on a finite-dimensional subcomplex.
Extremal metrics found on fibred Kähler manifolds with specific properties.
problem Finding constant scalar curvature Kähler metrics on fibred Kähler manifolds.
method Proving existence of extremal metrics under certain conditions on the base and fibres.
result Compact Kähler manifolds with specific properties admit constant scalar curvature Kähler metrics.
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…
We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…