Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.
problem Entropy norms and achirality of automorphisms on K3 and Enriques surfaces.
method Proves gap theorems for entropy norms and studies achirality in terms of genus-one fibrations.
result Entropy gaps and achirality results for automorphisms of K3 and Enriques surfaces.
Classifies and constructs all extendable automorphisms of closed surfaces over the 3-sphere.
problem Identifying extendable automorphisms of closed surfaces over the 3-sphere.
method Classification and construction of extendable automorphisms through embeddings and Heegaard surfaces.
result All extendable automorphisms of closed surfaces can be induced by automorphisms of the 3-sphere on Heegaard surfaces.
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.
Study shows pants graph automorphisms match mapping class groups of nonorientable surfaces.
problem Understanding automorphisms of pants graphs on nonorientable surfaces.
method Analyzing mapping class groups and proving isomorphism.
result Automorphism group of pants graphs isomorphic to mapping class groups.
Study automorphism groups of Inoue surfaces using quadratic number fields.
problem Understanding automorphism groups of Inoue surfaces.
method Construction and description of automorphism groups using quadratic number fields.
result Automorphism groups of Inoue surfaces S(+)/S(−) described in terms of quadratic number fields. Automorphisms of fine curve graph match surface homeomorphisms.
problem Understanding automorphisms of curve graphs for surfaces.
method Building on previous work, proving isomorphism to surface homeomorphisms.
result The group of automorphisms of the fine curve graph is isomorphic to the extended mapping class group of the surface.
The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.
problem Which elements of the mapping class group can be realized as affine automorphisms of dilation surfaces?
method Investigation into the affine automorphism groups of dilation surfaces, including the construction of dilation surfaces from multicurves.
result Only certain types of mapping class group elements can arise as affine automorphisms of dilation surfaces.
Study automorphisms of smooth curve graphs on surfaces.
problem Understanding automorphisms of fine curve graphs.
method Examined automorphisms of continuously differentiable curves on surfaces.
result Automorphisms on surfaces of genus ≥ 2 are induced by homeomorphisms.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
Automorphism group of nonorientable surface curve graph matches surface homeomorphisms.
problem Identifying automorphisms of nonorientable surface curve graphs.
method Using Bowden, Hensel, and Webb's fine curve graph and Long, Margalit, Pham, Verberne, and Yao's proof as a foundation.
result Automorphism group of nonorientable surface curve graph is isomorphic to the surface's homeomorphism group.
We determine explicitly the structure of the automorphism group of a parabolic Inoue surface. We also describe the quotients of the surface by typical cyclic subgroups of the automorphism group.
In this paper we introduce, for each closed orientable surface, an analogue of Tits buildings adjusted to investigation of the Torelli group of this surface. It is a simplicial complex with some additional structure. We call this complex with its additional structure the Torelli building of the surface in question. The…
Automorphisms of Kodaira surfaces are shown to be affine transformations.
problem Characterizing automorphisms of Kodaira surfaces.
method Analyzing lifts to the universal cover and conditions on affine transformations.
result Precise description of Kodaira surfaces' automorphism groups.
Automorphisms of fine 1-curve graph linked to surface homeomorphisms.
problem Understanding automorphisms of fine 1-curve graphs.
method Isomorphic mapping to surface homeomorphisms.
result Automorphism group is isomorphic to homeomorphism group of a surface.
Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.
problem Study of unlikely intersections for automorphisms of Markov surfaces with positive entropy.
method Arithmetic equidistribution for adelic line bundles, theory of laminar currents, quasi-Fuchsian representation theory.
result Two automorphisms with positive entropy share a Zariski dense set of periodic points if and only if they share a common iterate.
New examples of translation surfaces on hyperelliptic curves with many automorphisms.
problem Determining when translation surfaces are supported on the same algebraic curve.
method Analyzing eigenforms of automorphisms on hyperelliptic curves with many automorphisms.
result Presentation of infinitely many examples of translation surfaces on hyperelliptic curves.
Automorphisms of surfaces extendable over 4-sphere with invariant spin structures.
problem Extendability of automorphisms of surfaces over the 4-sphere.
method Constructing invariant spin structures and embeddings.
result Each automorphism of a surface is extendable over the 4-sphere after a connected sum with a torus.
Study automorphism equivariant Hitchin index for Riemann surfaces.
problem Define and study an index for Riemann surfaces under automorphisms.
method Define automorphism equivariant Hitchin index and prove a formula in terms of cohomological pairings.
result Prove a formula for the automorphism equivariant Hitchin index.
Study of flip graphs and their automorphism groups for infinite-type surfaces.
problem Understanding automorphism groups of flip graphs for infinite-type surfaces.
method Examined the relationship between mapping class groups and flip graphs for infinite-type surfaces.
result Extended mapping class groups are isomorphic to proper subgroups of automorphism groups of flip graphs.
Finite group actions on surfaces extend to 3-manifolds.
problem Classifying extendable periodic automorphisms of surfaces.
method Using fixed embeddings and 1-dominated 3-manifolds.
result All periodic automorphisms of closed surfaces extend to 3-spheres.
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.
New method computes automorphisms of surface groups using skein algebras.
problem Computing automorphisms of surface groups.
method Using skein algebras and Goldman Lie algebra.
result Refined formula for automorphisms of homology cylinders.
The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.
problem Presentations of mapping class groups of surfaces stabilizing boundaries.
method Gave presentations of mapping class groups of marked surfaces stabilizing boundaries.
result Presented cluster automorphism groups of cluster algebras from surfaces.
Automorphisms of fine graphs for surfaces and tori are studied.
problem Understanding automorphisms of fine graphs for surfaces and tori.
method Extending previous results to tori and discussing smooth versions.
result Automorphism groups of fine graphs for surfaces and tori are naturally isomorphic to homeomorphism groups.
No nontrivial automorphisms for cubic surfaces moduli space.
problem Understanding automorphisms of cubic surfaces moduli space.
method Analyzing the fundamental group of the moduli space.
result No nontrivial biholomorphic automorphisms for cubic surfaces moduli space.
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.
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.
Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S,f) to be conformally equivalent to (S',f') where S' is a bordered orientable Klein surface embedded in the Euclidean space and f' is the restrict…
Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…
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…
Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.
problem Understanding the dynamics of parabolic automorphisms on hyperkahler manifolds.
method Analyzing the action of parabolic automorphisms on the second cohomology group and fibers of Lagrangian fibrations.
result Parabolic automorphisms preserving Lagrangian fibrations act ergodically on the fibers.
This goal of the paper is to show that the automorphisms of the complex of curves in a surface are induced by the self-homeomorphisms of the surface except the surface is the 2-holed torus.
We show that the automorphism group of the complex of pants decompositions for a surface is isomorphic to the mapping class group for that surface.
Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.
problem Classifying periodic automorphisms on surfaces that commute with certain involutions.
method Analyzes irreducible periodic automorphisms on surfaces Σg that commute with hyperelliptic involutions. result A classification up to conjugacy for irreducible periodic automorphisms of a surface Σg commuting with involutions ι such that Σg/⟨ιangle is homeomorphic to T2. New rules for classifying certain square roots of surface automorphisms.
problem Classifying anticonformal square roots of automorphisms of even order on Riemann surfaces.
method Analyzing quotient surfaces and automorphism groups to determine topological equivalence.
result Conditions for equivalence of anticonformal square roots based on quotient surface genus and group structure.
Recently there has been renewed interest in the mapping-class group of a compact surface of genus g≥2 and also in its finite order elements. A finite order element of the mapping-class group will be a conformal automorphisms on some Riemann surface of genus g. Here we give the details of the proof that there is…
An automorphism f of a closed orientable surface Σ is said to be extendable over the 3-sphere S3 if f extends to an automorphism of the pair (S3,Σ) with respect to some embedding Σ↪S3. We prove that if an automorphism of a genus-2 surface Σ is extendable over S3, then f extends to …
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite ou…
The paper studies automorphisms of Riemann surfaces and their covers.
problem Understanding automorphisms of Riemann surfaces and their covers.
method Investigates the exact sequence of automorphisms and covers of Riemann surfaces.
result Determines conditions under which the exact sequence splits.
Study K3 surfaces and their metrics, focusing on dynamics.
problem Understanding dynamics on K3 surfaces.
method Interactions between K3 surface geometry and Ricci-flat metrics, dynamical study of automorphisms.
result Positive entropy automorphisms on K3 surfaces.
In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.
We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group C∗ associated to a compact connected Riemann surface.
Relation between generalized Weierstrass representation for conformal immersion of generic surfaces into three-dimensional space and Lax-Phillips scattering theory for automorphic functions is considered.
In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical …
The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. We characterize the mapping classes that restrict to periodic and reducible automorphisms of the surface.
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups o…
The first eigenvalue of the Laplacian on a unique Hurwitz surface has a sevenfold multiplicity and specific numerical values.
problem Identifying the spectrum of the Laplacian on a specific Hurwitz surface.
method Analytical proof for the multiplicity and numerical identification of the first eigenvalue; numerical identification of the eigenspace representation; determination of Dirichlet domain.
result The first eigenvalue of the Laplacian on the Fricke-Macbeath surface has a sevenfold multiplicity and is contained in the interval [1.23, 1.26].
The theorem connects surface mapping groups to fundamental groupoids.
problem Mapping class groups of bounded surfaces.
method Proving isomorphism between mapping class groups and fundamental groupoid automorphisms.
result Mapping class groups of bounded surfaces are isomorphic to fundamental groupoid automorphisms fixing boundary loops.