Related papers: Linear progress in the complex of curves
A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0)…
We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently:…
We study isometric actions of finitely presented groups on $\mathbb{R}$-trees. In this paper, we develop a relative version of the Rips machine to study $\textit{pairs}$ of such actions. An important example of a $\textit{pair}$ is a group…
For a random walk on the integer lattice $\mathbb{Z}$ that is attracted to a strictly stable process with index $\alpha\in (1, 2)$ we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set.…
The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the…
In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the…
We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also…
We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its…
Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still…
We propose a simple, geometrically-motivated construction of smooth random paths in the plane. The construction is such that, with probability one, the paths have finite curvature everywhere (and the realizations are visually pleasing when…
We prove that there exist infinitely many asymptotics of drift for random walks on finitely generated groups.
We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant…
A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the `translation distance' of the…
We consider the class of quasiprojective varieties admitting a dominant morphism onto a curve with negative Euler characteristic. The existence of such a morphism is a property of the fundamental group. We show that for a variety in this…
We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either…
Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow…
In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the…
We investigate a space-inhomogeneous discrete-time quantum walk in one dimension. We show that the walk exhibits localization by a path counting method.
This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being…
We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…