arXiv research

This paper shows how every group can be realized as automorphisms of a dessin d'enfant.

Study automorphisms of pure braid groups on sphere homotopy groups.

We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…

Study automorphism group actions on Jacobi diagrams spaces.

Study automorphism groups' action on Jacobi diagrams, leading to new decompositions.

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.

For n>2, the action of the outer automorphism group of the rank n free group F_n on the SU(2)-character variety Hom(F_n,SU(2))/SU(2)$ is ergodic with respect to the Lebesgue measure class.

We prove that the automorphism group of a compact 6-manifold $M$ endowed with a symplectic half-flat SU(3)-structure has abelian Lie algebra with dimension bounded by min$\{5,b_1(M)\}$. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new com…

We show that the isotropy action of a homogeneous space $G/K$, where $G$ and $K$ are compact, connected Lie groups and $K$ is defined by an automorphism on $G$, is equivariantly formal and that $(G, K)$ is a Cartan pair.

The earthquake flow is asymmetric and cannot be extended to an SL(2,R) action.

Free groups' automorphisms have bounded orbits.

Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.

The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polyno…

Finite group actions on surfaces extend to 3-manifolds.

Origami curves link surface automorphisms to group actions.

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfie…

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…

We show that tree almost automorphism groups, including Neretin groups, satisfy the analogue of the $F_\infty$-finiteness condition in the world of totally disconnected groups: They possess a cellular action on a contractible cellular complex such that the stabilizers are open and compact and the restriction of the act…

The paper studies the action of automorphisms on train tracks and finds that the set of minimally displaced points is co-compact.

We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\T^N$.

Vanishing of equivariant cohomology groups for proper Lie group actions.

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

Compact groups can be represented as dessin automorphisms.

Compact Lie groups can be realized as automorphism groups of Riemannian manifolds.

Study mapping class group actions on surface fundamental groups.

Study automorphisms on procongruence curve and pants complexes.

Finite index subgroups of certain groups cannot act faithfully on the circle.

Classifies toroidal circle planes with 3D automorphism groups.

Study automorphism equivariant Hitchin index for Riemann surfaces.

Study of special subgroups of automorphism groups of Kronrod-Reeb graphs for Morse functions on 2-torus.

We extend Forester's rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups).

We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an aut…

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Pro…

Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.

In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group $G$. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of $G$. We prove a result which allows us to establish the same conclusion when $G$ is assumed …

New examples of translation surfaces on hyperelliptic curves with many automorphisms.

Graph theory connects automorphisms to cohomology.

In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector …

Introduces group-valued momentum maps for symplectic fiber bundles.

Detects free group automorphisms using homology of covers.

Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.

Let $M^{r}$ be a connected orientable manifold with the Euler characteristic $χ(M)\not \equiv 0\operatorname{mod}6$. Denote by $\mathrm{SAut}(F_{n})$ the unique subgroup of index two in the automorphism group of a free group. Then any group action of $\mathrm{SAut}(F_{n})$ (and thus the special linear group $\mathrm{SL…

We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a …

We show that the fixed point set of a proper action of a Lie group $G$ on a Poisson manifold $M$ by Poisson automorphisms has a natural induced Poisson structure and we give several applications.

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

We describe the action of the automorphism group of the complex cubic x^2+y^2+z^2-xyz-2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2,C) character variety given by fixing the trace of the peripheral element (so-called "rela…