Tree almost automorphism groups have a cellular action on a contractible complex.
problem Understanding the structure of tree almost automorphism groups.
method Showed cellular action on a contractible complex with specific stabilizers.
result Tree almost automorphism groups satisfy the F∞-finiteness condition. New groups constructed from tree automorphisms, proving finiteness.
problem Finiteness properties of self-similar and Röver groups.
method Braiding of self-similar groups, use of d-ary cloning systems, analysis of embedded disks. result Construction of a new group (braided Röver group) of type F∞. Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding …
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.
Study of tree automorphisms via arc-stabilizers.
problem Understanding automorphisms of R-trees. method Analyzing actions over arc-stabilizers.
result Point-stabilizers are finitely generated.
We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.
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.
We consider certain groups of tree automorphisms as so-called diffeological groups. The notion of diffeology, due to Souriau, allows to endow non-manifold topological spaces, such as regular trees that we look at, with a kind of a differentiable structure that in many ways is close to that of a smooth manifold; a suita…
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.
Introduces a theorem for groups acting on trees.
problem Understanding group actions on trees.
method Two proofs of a Small Cancellation Theorem.
result Application to plane polynomial automorphisms.
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…
The paper explores 6D almost complex structures with maximal symmetry groups.
problem Finding the largest automorphism groups of 6D almost complex structures.
method Analyzing the Nijenhuis tensor and automorphism groups of 6D almost complex structures.
result The largest automorphism group dimension is 10, realized by 3 strictly nearly Kähler spaces.
Vanishing cup product of Brooks quasimorphisms proved.
problem Vanishing cup product of Brooks quasimorphisms.
method Vanishing of the square of a universal class for tree automorphism groups.
result Vanishing cup product of Brooks quasimorphisms.
For a fully irreducible automorphism φof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'φ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'φ^n for n large.
This work extends lamination theory to free products, describing Gromov boundaries and subgroup classification.
problem Classifying subgroups of outer automorphisms of free products.
method Extending lamination theory to free products, using Rips machine and Rauzy-Veech induction.
result A 2-to-1 map from the boundary of the group to an R-tree, with unique duality for arational trees. We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric.…
Study of automorphisms and splittings of special groups, showing infinite groups under certain conditions.
problem Understanding the structure and automorphisms of special groups G. method Constructing and analyzing non-small, stable G-actions on R-trees. result Conditions for the existence of infinite-order automorphisms and splittings.
Random subgroups of free groups are invariant only by inner automorphisms.
problem Understanding the structure of random subgroups in free groups.
method Analyzing splittings and automorphisms of random subgroups.
result Random subgroups of free groups are invariant only by inner automorphisms.
Vanishing of cohomology for SL_2 groups over special fields.
problem Vanishing of cohomology for SL_2 groups over non-Archimedean local fields and S-integers.
method Analyzing continuous bounded cohomology of SL_2(k) and generalizing to tree automorphisms.
result Continuous bounded cohomology of SL_2(k) vanishes in all positive degrees for non-Archimedean local fields k.
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.
Spaces of circle embeddings in curved surfaces indexed by trees.
problem Classifying spaces of braided automorphism groups of trees.
method Indexed connected components with finite rooted trees, constructed strong deformation retract.
result Connected components are classifying spaces of braided automorphism groups.
For a finitely generated group G, we introduce an asymmetric pseudometric on projectivized deformation spaces of G-trees, using stretching factors of G-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…
Classifies toroidal circle planes with 3D automorphism groups.
problem Classifying toroidal circle planes with specific automorphism groups.
method Using almost simple Lie groups and their isomorphism to PSL(2, R), a framework for classification is described.
result Three-dimensional connected automorphism groups are isomorphic to PSL(2, R).
Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.
problem Understanding bounded cohomology of groups with prescribed local actions.
method Proving vanishing or infinite bounded cohomology based on the 2-transitivity of F′. result Vanishing or infinite bounded cohomology depending on F′'s 2-transitivity. We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain …
The paper classifies isometries on specific Lie groups.
problem Classifying isometries on nonnilpotent, solvable 3D Lie groups.
method Proving automorphisms are the only isometries for rank two Almost-Riemannian Structures.
result A classification result for rank two ARSs on nonnilpotent, solvable 3D Lie groups.
The paper describes automorphism groups and limit sets for finite type bounded domains.
problem Characterizing automorphism groups and limit sets for finite type bounded domains.
method Detailed analysis of automorphism groups and limit sets for pseudoconvex domains with finite type.
result The automorphism group has finitely many components and is the almost direct product of a compact group and a connected Lie group.
If G is a free product of finite groups, let ΣAut1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] an…
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. We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familliar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.
For convex domains, automorphism group and limit set properties are described.
problem Characterize the automorphism group and limit set of convex domains.
method Detailed analysis of automorphism group structure and limit set properties for convex domains with C1,ε boundary. result The automorphism group has finitely many components and the limit set is homeomorphic to a sphere.
The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.
problem Representing automorphisms of hyperbolic groups using train track maps.
method Using graphs of groups and Bestvina-Handel's irreducible train track maps, the thesis constructs relative train track maps.
result Outer automorphisms of finitely-generated word hyperbolic groups satisfy a dynamical trichotomy.
Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depe…
New structures with symmetry found, contradicting previous assumptions.
problem Limitations of C1 regularity in almost-Grassmannian structures. method Constructing families of (2,n)-almost Grassmannian structures with C1 regularity and specific symmetry. result Theorem 1.3 of [9] is not valid under C1 regularity assumptions. Study automorphisms of complex foliations in 3D.
problem Computing automorphisms of complex foliations in specific cases.
method Analyzing automorphisms of 3D Reeb components obtained by Hopf construction.
result Almost complete description of leafwise holomorphic automorphisms.
6D manifolds with special symmetries have limited automorphism groups.
problem Understanding the maximum symmetry groups of 6D manifolds.
method Analyzing almost product structures and their automorphism groups.
result The automorphism group of 6D manifolds with non-degenerate Nijenhuis tensor has a limited dimension, with the maximum being 14.
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).
Study the boundary of hyperbolic groups generated by atoroidal automorphisms.
problem Characterize the Gromov boundary of hyperbolic groups generated by atoroidal automorphisms.
method Define directional Whitehead graphs and prove properties of indecomposable trees. Use these to show boundary homeomorphism to Menger curve.
result The boundary of hyperbolic groups generated by atoroidal, fully irreducible automorphisms is homeomorphic to the Menger curve.
This paper and its companion arXiv:0911.3173 have been replaced by arXiv:1602.05139. We define the compatibility JSJ tree of a group G over a class of subgroups. It exists whenever G is finitely presented and leads to a canonical tree (not a deformation space) which is invariant under automorphisms. Under acylindricity…
We find canonical decompositions for finitely presented groups which specialize to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbou…
Theory of structure trees applied to network max-flow min-cut theorem and group theory.
problem Max-Flow Min-Cut Theorem and its generalizations for infinite networks and groups.
method Theory of structure trees and edge cuts in networks.
result Generalization of Max-Flow Min-Cut Theorem to infinite networks and new insights into group structure.
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.
Study of quadratic form associated with surface automorphisms and its applications to singularity theory.
problem Understanding the properties of quadratic forms associated with surface automorphisms and their applications to singularity theory.
method Using techniques from mapping class group theory, the authors associate a quadratic form and prove its properties using the twist formula.
result The form ildeQ is positive definite under certain conditions and even in others, providing numerical invariants to distinguish different topological types of singularities. Study the boundary of Riemann surfaces with abelian automorphisms.
problem Characterize the boundary of Riemann surfaces with abelian automorphisms.
method Analyze the moduli space and its Deligne-Mumford compactification, focusing on equisymmetric loci.
result Describe the topological strata at the boundary for hyperelliptic and cyclic p-gonal actions. We study cocompact lattices with dense projections in a product G1×G2 of locally compact groups and show, under the assumption that each Gi is a closed subgroup of the automorphism group Aut(Ti) of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only…
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.
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.
The study proves super-rigidity of Gromov's random monster group for various types of groups.
problem Super-rigidity of Gromov's random monster group in various group types.
method Proof of morphisms having finite image and introduction of hereditary super-rigidity.
result Gromov's random monster group has super-rigidity and hereditary super-rigidity with respect to certain groups.