arXiv research

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…

Homotopy equivalent boundaries of cube complexes are studied.

The Tits alternative applies to groups acting on specific CAT(0) spaces.

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…

New rigidity result for CAT(0) spaces of higher rank.

We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theor…

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish tha…

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…

We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preservin…

We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that…

Uniform proof reconstructs spaces using cross ratio on boundary.

Proves strong Tits alternative for 3D automorphisms over zero char fields.

Study of Coxeter diagrams and Artin-Tits groups, focusing on normalisers and wall intersections.

Study the Gromov boundary of fine curve graph for surface homeomorphisms.

We study the asymptotic cones of the universal covering spaces of closed 4-dimensional nonpositively curved real analytic manifolds. We show that the existence of nonstandard components in the Tits boundary, discovered by Christoph Hummel and Victor Schroeder, depends only on the quasi-isometry type of the fundamental …

We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete,…

Study growth rates of automorphisms of special groups.

The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.

Uniform proof of property R_infinity for specific Artin-Tits groups.

We prove that an Artin-Tits group of type $\tilde C$ 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 …

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

Characterizes periodic elements in Artin-Tits groups via stability conditions.

Study on faithfulness of Burau representation for Artin-Tits groups.

We study affine maps between CAT(0) spaces with geometric actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with geometric …

Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mappi…

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 $n \geq 4$, the $n$-st…

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy t…

The Tits alternative for Out(F_n) is reduced to the case where all elements in the subgroup under consideration grow polynomially.

Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that the boundary of X has no cut points and that one can detect splittings of $G$ over two-ended groups and recover its JSJ decomposition from the boundary. We show that any discrete action of a group G on a CAT(0) space X satis…

Using the construction of a nonorientable Curtis-Tits group of type $\tilde A_n$, we obtain new explicit families of expander graphs of valency five for unitary groups over finite fields.

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry i…

Study shows similar result to Margulis for Cantor set homeomorphisms.

Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always contains a rank 2 free group.

Adapts pivoting technique to circle homeomorphisms for proofs.

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…

It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using…

New proofs for growth series of Coxeter groups using complex structures.

In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely many primes), including torsion-freeness, a Tits alternative, and …

The proof of the Tits alternative for $Out(F_n)$ is completed. The main tool is a Kolchin type theorem, proved in this paper. It states that a finitely generated subgroup of $Out(F_n)$ consisting of unipotent automorphisms can be conjugated into an upper-triangular subgroup (this is interpreted via train-tracks).

In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type $\tilde A_n$ 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…

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting fe…

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…

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…

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semi-algebraic subsets of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary was …

Study Coxeter groups over fusion rings and their geometric realisations.