Proves strong Tits alternative for 3D automorphisms over zero char fields.
problem Tits alternative for 3D tame automorphisms over zero char fields.
method Proved strong Tits alternative using 3D tame automorphisms over zero char fields.
result Strong Tits alternative proven for 3D tame automorphisms over zero char fields.
Study of Coxeter diagrams and Artin-Tits groups, focusing on normalisers and wall intersections.
problem Understanding normalisers of parabolic subgroups in Artin-Tits groups and their connections to Coxeter diagrams.
method Analyzing hyperplane arrangements, Coxeter groups, and wall-and-chamber structures.
result Complexified hyperplane complement is a K(π,1) space for normalisers of parabolic subgroups in finite-type Coxeter diagrams.
The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.
problem Defining and studying a Tits building for commutative rings.
method Proving a Solomon-Tits theorem for commutative rings under specific conditions, defining Steinberg modules, and computing ranks and lengths.
result Proves a Solomon-Tits theorem for commutative rings satisfying certain conditions.
Proves Tits alternative for specific PD(3) groups.
problem Tits alternative for PD(3) groups. method Proving Tits alternative for almost coherent PD(3) groups. result Almost coherent PD(3) groups contain rank 2 free groups. We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary t…
Artin-Tits groups act loxodromically on a delta-hyperbolic complex, with most elements doing so.
problem Understanding the proportion of elements in Artin-Tits groups that act loxodromically.
method Analogy with mapping class groups and study of the additional length complex.
result Most elements of Artin-Tits groups act loxodromically, with a condition guaranteeing this proportion stays away from zero.
Uniform proof of property R_infinity for specific Artin-Tits groups.
problem Establishing property R_infinity for various Artin-Tits groups.
method Uniform short proof for multiple groups using similar techniques.
result Property R_infinity confirmed for specified Artin-Tits groups.
We prove that an Artin-Tits group of type C~ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives …
Proves Tits alternative for finite rank median spaces, restricting group actions.
problem Understanding group actions on median spaces.
method Extending Caprace-Sageev machinery and Hagen's theory to median spaces.
result Groups acting on finite rank median spaces are more restricted.
Paper discusses hyperbolic structures for Artin-Tits groups.
problem No specific problem stated, focuses on hyperbolic structures.
method Algebraic analogues of previously known structures on Artin braid groups.
result Presented several candidates for hyperbolic structures.
Artin-Tits groups of spherical type have parabolic subgroups with lattice properties.
problem Characterize parabolic subgroups in Artin-Tits groups of spherical type.
method Prove intersection and inclusion properties, show minimal parabolic subgroups, and define a simplicial complex.
result Parabolic subgroups form a lattice and have unique minimal subgroups.
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
Study on faithfulness of Burau representation for Artin-Tits groups.
problem Determining faithfulness of Burau representation for Artin-Tits groups.
method Algorithmic approach based on categorical methods.
result Burau representation is not faithful in specific cases.
The Tits alternative applies to groups acting on specific CAT(0) spaces.
problem Understanding the structure of groups acting on CAT(0) spaces.
method Proving the Tits alternative for groups acting on visibility CAT(0) spaces with bounded packing property.
result Groups either almost nilpotent or contain a free nonabelian subgroup of rank 2.
The paper explores centers of subgroups in mapping class groups and their relation to free groups and Tits alternatives.
problem Investigating centers of subgroups in mapping class groups and their properties.
method Similar techniques to show the presence of nonabelian free groups and failure of Tits alternatives.
result No big mapping class group satisfies the strong Tits alternative, and many have trivial centers.
We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-dimensional type. A particular case is that for n≥4, the n-st…
Algorithm finds minimal standardizer of parabolic subgroups in Artin-Tits groups.
problem Computing minimal standardizer of parabolic subgroups in Artin-Tits groups.
method Geometrical and algebraic algorithms to compute pn-normal form of a central element. result Minimal standardizer computation simplified to pn-normal form. Groups act freely on CAT(0) cube complexes, with strong Tits alternative properties.
problem Understanding the action of tubular free by cyclic groups on CAT(0) cube complexes.
method Using Wise's equitable sets criterion and properties of CAT(0) cube complexes.
result Tubular free by cyclic groups have finite index subgroups satisfying the strongest Tits alternative.
Let G=G1∗⋯∗Gk∗F be a countable group which splits as a free product, where all groups Gi are freely indecomposable and not isomorphic to Z, and F is a finitely generated free group. If for all i∈{1,…,k}, both Gi and its outer automorphism group Out(Gi) satisfy t…
The Tits alternative for Out(F_n) is reduced to the case where all elements in the subgroup under consideration grow polynomially.
Using the construction of a nonorientable Curtis-Tits group of type A~n, we obtain new explicit families of expander graphs of valency five for unitary groups over finite fields.
Study shows similar result to Margulis for Cantor set homeomorphisms.
problem Understanding groups of homeomorphisms of Cantor sets.
method Analogous to Margulis's proof for linear groups.
result Groups of homeomorphisms either preserve a measure or contain a free subgroup.
Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.
problem Computing reduction systems in Artin-Tits groups of spherical type.
method Introduces a canonical reduction system, proves periodicity of centralizers, and provides algorithms.
result Improved algorithms for computing reduction systems in braid groups and Artin-Tits groups.
The paper proves a quantitative Tits alternative for negatively pinched manifolds.
problem Proving a quantitative version of the Tits alternative for negatively pinched manifolds.
method Analyzing discrete isometry subgroups generated by two non-elliptic isometries.
result A free subgroup of rank 2 is found in the isometry subgroup, which is convex-cocompact when one of the generators is hyperbolic.
Adapts pivoting technique to circle homeomorphisms for proofs.
problem Probabilistic Tits alternative and exponential synchronization.
method Adapts Gou{ë}zel's pivoting technique.
result Different proofs of probabilistic Tits alternative and exponential synchronization.
There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical bui…
New proofs for growth series of Coxeter groups using complex structures.
problem Proving new formulae for growth series of Coxeter groups.
method Using the structure of Coxeter complexes, Davis complexes, or Tits non-complexes.
result Several classical formulae for growth series are proved in a new way.
In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type A~n we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter…
The proof of the Tits alternative for Out(Fn) is completed. The main tool is a Kolchin type theorem, proved in this paper. It states that a finitely generated subgroup of Out(Fn) consisting of unipotent automorphisms can be conjugated into an upper-triangular subgroup (this is interpreted via train-tracks).
New method calculates growth rates of Artin-Tits monoids, linking to partial theta function.
problem Growth rates of Artin-Tits monoids and their connection to partial theta function.
method Determining growth functions using simple matrix determinants and atomic generators.
result Exponential growth rates of Artin-Tits monoids of type An tend to 3.233636... as n increases. We construct a family of morphisms between Artin-Tits groups which generalise the ones constructed by J. Crisp in [Injective maps between Artin groups, Proceedings of the Special Year in Geometric Group Theory, Berlin, (1999), 119 -- 138]. We show that their restrictions to the positive Artin monoids respect normal for…
Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geode…
Study Coxeter groups over fusion rings and their geometric realisations.
problem Understanding Coxeter groups and their embeddings.
method Investigate faithful realisations and Vinberg systems.
result Induce embeddings of hyperplane complements.
The paper shows that certain manifolds have a dimension less than half.
problem Understanding the geometric structure of manifolds with nonpositive curvature.
method Constructing a geometric analog of the Tits building and proving dimensionality.
result The dimension of the analog is less than half the dimension of the manifold.
We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparam…
Combinatorial proof confirms two exceptional compact Tits geometries of type C3 are simply connected.
problem Proving two exceptional compact Tits geometries of type C3 are simply connected.
method Combinatorial proof independent of Kramer and Lytchak's result.
result Two exceptional compact Tits geometries of type C3 are simply connected.
New rigidity result for CAT(0) spaces of higher rank.
problem Rigidity of CAT(0) spaces of rank at least 2.
method Study of Morse flats and their Tits boundaries.
result CAT(0) spaces of rank at least n contain a regular point in their Tits boundary.
New approach to parabolic subalgebras and buildings.
problem Understanding the geometry of parabolic subalgebras and buildings.
method Elementary approach over arbitrary fields, focusing on parabolic subalgebras and their properties.
result Derivation of structure theory from root systems to Bruhat decomposition.
Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…
Groups acting on CAT(0) spaces without 3-flats have rigid properties.
problem Understanding group actions on CAT(0) spaces without 3-flats.
method Analyzing the structure of CAT(0) spaces and group actions.
result Groups acting geometrically on such spaces have rigid properties.
New proof of surface group theorem for 2D Poincaré duality groups.
problem Characterizing groups with specific algebraic properties.
method Analyzing amenability and homological isoperimetric inequalities.
result Groups satisfying certain conditions are either amenable or have linear homological isoperimetric inequalities.
We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that G is a group for which there is a bound on the orders of its finite subgroups. We prove that if G acts properly on a finite-dimensional CAT(0) cubical complex, then either G contains a free subgroup of rank 2 o…
Defines connections on parahoric torsors for curves and proves their equivalence to polystability.
problem Defining and characterizing connections on parahoric torsors over curves.
method Definition of parahoric $\cG$--torsors and connections, proving equivalence to polystability.
result A $\cG$--torsor is polystable if and only if it is given by a homomorphism to a maximal compact subgroup.
CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.
problem Understanding geometric properties of CAT(0) spaces near spheres.
method Analyzing the relationship between CAT(0) spaces and their Tits boundaries.
result CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.
Introduces Morse quasiflats and proves their equivalence and quasi-isometry invariance.
problem Generalizing Morse quasigeodesics to arbitrary dimensions.
method Introduces alternative definitions and proves their equivalence under appropriate assumptions.
result Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries.
We classify isometries of compact Lorentz manifolds.
problem Understanding the isometry group of compact Lorentz manifolds.
method Proving structure theorems and applying the Tits alternative.
result Classification of lattices acting on compact Lorentz manifolds.
We obtain an analog of the compression of angles theorem in symmetric spaces for Bruhat--Tits buildings of the type A. More precisely, consider a p-adic linear space V and the set Lat(V) of all lattices in V. The complex distance in Lat(V) is a complete system of invariants of a pair of points of Lat(V) u…