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The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the…
We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then…
We prove that for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.
We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance
We present a procedure of group cubization: It results in a group whose some features resemble the ones of a given group, and which acts without fixed points on a CAT(0) cubical complex. As a main application we establish lack of Kazhdan's…
This paper is concerned with some Algebraic Geometry codes on Jacobians of genus 2 curves. We derive a lower bound for the minimum distance of these codes from an upper "Weil type" bound for the number of rational points on irreducible…
Our goal is to develop a limit approach for a class of problems in additive combinatorics that is analogous to the limit theory of dense graph sequences. We introduce metric, convergence and limit objects for functions on groups and for…
In this article, we study subloci of solvable curves in $\mathcal{M}_g$ which are contained in either a K3-surface or a quadric or a cubic surface. We give a bound on the dimension of such subloci. In the case of complete intersection genus…
We give a characterization of limits of dihedral groups in the space of finitely generated marked groups. We also describe the topological closure of dihedral groups in the space of marked groups on a fixed number of generators.
The projective linear group \(\pgl(\comp,4)\) acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a…
In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.
For locally finite CAT(0) cube complexes it is known that they are injectively metrizable choosing the $l_\infty$-norm on each cube. In this paper we show that cube complexes which are injective with respect to this metric are always…
We revisit the topic of probability measures on CAT(0) cube complexes and prove that an amenable group acting on a CAT(0) cube complex, regardless of dimension, necessarily preserves an interval in the Roller compactification. In the finite…
It is proved that the mapping class group of any closed surface with finitely many marked points is quasiisometric to a CAT(0) cube complex. We provide two distinct proofs, one tailored to mapping class groups, and one applying to a larger…
We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a CAT(0) cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and…
We study linearizability of actions of finite groups on cubic threefolds with nonnodal isolated singularities.
We provide a condition on the links of polygonal complexes that is sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes contain a virtually free codimension-1 subgroup. We provide stronger conditions…
Sufficient conditions are given for the computation of accessing arcs and arcs that links boundary components of multiply connected domains. The existence of a not-computably-accessible but computable point on a computably compact arc is…
Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…
We consider finite 2-dimensional polyhedral complexes, equipped with piecewise non-positively curved, locally CAT(0) metrics. We give conditions on the complex X that ensure that its fundamental group contains a surface subgroup. Concrete…