Study on endomorphism and automorphism groups of specific quandles.
problem Characterizing endomorphism and automorphism groups of residually finite and profinite quandles.
method Proved properties of endomorphism monoids and automorphism groups for residually finite and profinite quandles.
result Endomorphism and automorphism groups of residually finite quandles are residually finite.
Automorphism groups of Hopf manifolds are finite and have a bounded order.
problem Understanding the structure of automorphism groups of Hopf manifolds.
method Proved that the automorphism groups of Hopf manifolds are Jordan.
result Automorphism groups of Hopf manifolds are finite and have a bounded order.
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.
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.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.
problem Finiteness properties and contractibility of symmetric automorphisms of RAAGs.
method Definition of symmetric automorphism group, construction of symmetric Outer space, proof of contractibility.
result Finiteness properties and contractibility results for symmetric automorphisms of RAAGs.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We…
Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.
problem Characterizing the basic automorphism groups of Cartan foliations.
method Analyzes sufficient conditions and estimates dimensions for basic automorphism groups of Cartan foliations covered by fibrations.
result Identifies sufficient conditions for the existence of a finite-dimensional Lie group structure in basic automorphism groups.
Finite automorphisms of SL(2,C) character varieties identified.
problem Character varieties of surfaces with negative Euler characteristic.
method Algebraic automorphisms and mapping class groups studied.
result Finite extension of mapping class group identified.
An automorphism α of a group G is normal if it fixes every normal subgroup of G setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group G, Inn(G) has finite index in the subgroup Autn(G) of normal au…
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.
Outer automorphisms of free products are represented by CTs.
problem Representing outer automorphisms of free products by relative train track maps.
method Developed theory of attracting laminations and CTs for free products.
result Outer automorphisms of free products satisfy an index inequality.
Study of infinite-type surfaces' automorphisms and graph structures.
problem Understanding automorphisms of infinite-type surfaces.
method Isomorphic mappings between extended mapping class groups and graph automorphism groups.
result Extended mapping class groups are isomorphic to graph automorphism groups.
We embed arbitrary groups into regular graphs with prescribed automorphisms.
problem Embedding arbitrary groups into regular graphs with specific automorphisms.
method Constructing regular graphs with strong embeddings and automorphism groups isomorphic to any given finite group.
result For every d≥3 and every finite group G, there exists a d-regular graph Γ with a strong embedding β such that Aut(Γ)≅Aut(β(Γ))≅G. 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 study examines groups with a specific automorphism property using BNS-invariant.
problem When does a group have the finitely generated fixed subgroup property?
method Using BNS-invariant to analyze groups of the form GimesZm. result Provides partial answers and examples for groups with the property.
Finite groups can be represented as origami automorphisms, extended to countable groups.
problem Representing countable groups as automorphisms of origamis.
method Considering origamis on the Loch Ness monster.
result Every countable group can be represented as origami automorphisms.
The study examines how automorphism growth rates of a group can be deduced from its simpler decompositions.
problem Determine automorphism growth rates of a group from its simpler decompositions.
method Analyze group decompositions into simpler pieces (direct products, free products, graph of groups) and deduce growth rates.
result Information about automorphism growth rates of a group can be deduced from its simpler decompositions.
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…
For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and …
Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
problem Characterizing the hyperbolicity of automorphism groups of infinitely-ended groups.
method Proving acylindrical hyperbolicity through group properties and automorphism analysis.
result Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
Automorphisms and subdivisions of Helly graphs are studied, leading to explicit models and rational translation lengths.
problem Understanding automorphisms and subdivisions of Helly graphs.
method Simple fine simplicial subdivisions and explicit simplicial models of the injective hull.
result Any automorphism of a Helly graph is either elliptic or hyperbolic, with rational translation lengths.
We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.
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∞. Finite groups of RAAGs act on a contractible space without fixed points.
problem Characterizing finite subgroups of RAAG outer automorphisms.
method Analyzing the action of finite subgroups on a contractible space.
result Finite subgroups of RAAG outer automorphisms fix a point in the space.
Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to…
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.
We study the outer automorphism group of a right-angled Artin group AΓ with finite defining graph Γ. We construct a subnormal series for Out(AΓ) such that each consecutive quotient is either finite, free-abelian, GL(n,Z), or a Fouxe-Rabinovitch group. The last two types act respectively on a symmetri…
The paper studies groups of surface automorphisms and their representations.
problem Understanding the structure of groups of surface automorphisms and their representations.
method Analyzes group homomorphisms between Diff+(S) and Sp(H1(S)). result Provides conditions for the image of the homomorphism to have finite index or no finite nonzero orbit.
Study growth rates of automorphisms of special groups.
problem Understanding the growth rates of automorphisms of special groups.
method Analyzing outer automorphisms of virtually special groups, showing polynomial or exponential growth, and constructing Nielsen-Thurston decompositions.
result Outer automorphism groups of virtually special groups are boundary amenable, have finite virtual cohomological dimension, and satisfy the Tits alternative.
The automorphisms of free groups with boundaries form a family of groups A_{n,k} closely related to mapping class groups, with the standard automorphisms of free groups as A_{n,0} and (essentially) the symmetric automorphisms of free groups as A_{0,k}. We construct a contractible space L_{n,k} on which A_{n,k} acts wit…
Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
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.
Normal closures of certain finite subgroups in automorphism and outer automorphism groups of free groups are determined.
problem Determining the normal closures of specific finite subgroups in automorphism and outer automorphism groups of free groups.
method Analyzing the structure of finite subgroups and their normal closures in Aut(Fn) and Out(Fn). result The normal closure of certain finite subgroups in Aut(Fn) and Out(Fn) are determined. Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.
problem Understanding the automorphisms of sphere complexes associated with graphs.
method Analyzing the group of proper homotopy equivalences and constructing an exhaustion of the sphere complex.
result The automorphism group of the sphere complex is isomorphic to the mapping group of the graph.
New proof shows maximal arithmetic groups are finite.
problem Finiteness of maximal arithmetic reflection groups.
method Arithmetic Margulis lemma without automorphic forms.
result Finiteness of maximal arithmetic reflection groups proven.
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.
Study shows automorphism groups of certain hyperbolic manifolds are infinitely generated.
problem Understanding the homotopy groups of automorphism groups of hyperbolic manifolds.
method Analyzing the homotopy groups of specific topological groups and applying these results to automorphism groups of hyperbolic manifolds.
result The (n-4)-th homotopy group of the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (n >= 4) is infinitely generated.
Let G be a finitely generated polyfree group. If G has nonzero Euler characteristic then we show that Aut(G) has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain G of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We…
Let G be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from G to Z have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing G as the mapping torus of a free grou…
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…
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
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.
In the present paper, we show that many combinatorial and topological objects, such as maps, hypermaps, three-dimensional pavings, constellations and branched coverings of the two--sphere admit any given finite automorphism group. This enhances the already known results by Frucht, Cori -- Machì, Širáň -- Škoviera, and …
Generators found for automorphisms of special groups.
problem Characterizing automorphisms of special groups.
method Using a novel hierarchical shortening argument, the paper provides generators for automorphisms of special groups.
result Generators for automorphisms of special groups are found, including Dehn twists and pseudo-twists.
Residually finite groups found in manifold automorphisms.
problem Residual finiteness of automorphism groups of high-dimensional manifolds.
method Embedding calculus, Weiss fibre sequence, convergence of embedding calculus tower, smoothing theory.
result Topological mapping class group of high-dimensional manifolds is residually finite.
In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a δ-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.