arXiv subjects
Publications and source records attributed to Aitor Azemar.
We prove that the set of Busemann points (the limits of almost-geodesic rays) is nowhere dense in the horoboundary of the Teichm\"uller metric for all Teichm\"uller spaces of complex dimension strictly larger than 1. This shows that the Teichm\"uller metric is far from having non-positive curvature in a certain sense.
Stationary measures on the circle that arise from a large class of random walks on the fundamental group of a finite-area complete hyperbolic surface with cusps are singular with respect to the Lebesgue measure. In particular, it is sufficient for singularity that a stationary measure satisfies an exponential decay for cusp excursions with excursions being measured in the path metric on horocycles bounding cusps. In this note, we settle a conjecture of McMullen by proving that the singularity of stationary measures satisfying such exponential decay is quasi-symmetrically stable, that is under push-forward by any quasi-symmetry of the circle the measure remains singular.
Consider a closed surface $S$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $S$ with finite first moment with respect to some hyperbolic metric on $S$. Corresponding to each point in Teichm\"uller space there is an associated random walk on the hyperbolic plane. Azemar--Gadre--Gou\"ezel--Haettel--Lessa--Uyanik prove that the drift of this random walk is a proper function on Teichm\"uller space, and that this drift grows at least linearly with respect to the Teichm\"uller distance. In this paper we refine the result. On the one hand, by considering Jenkins-Strebel directions we show that the linear lower bound is sharp. On the other hand, we show that for Lebesgue typical Teichm\"uller geodesics, the drift grows exponentially. We also exhibit Teichm\"uller geodesics for which the growth oscillates between almost linear and exponential. Furthermore, we show that the drift is a quasiconvex function up to a multiplicative constant.
Consider a closed surface $M$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $M$ with finite first moment. Corresponding to each point in the Teichm\"uller space of $M$, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichm\"uller space of $M$, and we relate the growth of the speed to the Teichm\"uller distance to a basepoint. One key argument is an adaptation of Gou\"ezel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.
Two commonly studied compactifications of Teichm\"uller spaces of finite type surfaces with respect to the Teichm\"uller metric are the horofunction and visual compactifications. We show that these two compactifications are related, by proving that the horofunction compactification is finer than the visual compactification. This allows us to use the simplicity of the visual compactification to obtain topological properties of the horofunction compactification. Among other things, we show that the horoboundary of Teichm\"uller space is path connected and that its Busemann points are not dense, we determine for which surfaces the horofunction compactification is isomorphic to the visual one, and we show that some horocycles diverge in the visual compactification based at some point. As an ingredient in one of the proofs we show that extremal length is not $C^{2}$ along some paths that are smooth with respect to the piecewise linear structure on measured foliations.
We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moment with respect to the Teichmuller metric, and whose supports generate non-elementary subgroups. We prove that Teichmuller space with the Teichmuller metric is statistically hyperbolic for such a harmonic measure.
We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular we prove that if a convergence group $G$ acts on a compact metrizable space $M$ with the convergence property then we can provide $G\cup M$ with a compact topology such that random walks on $G$ converge almost surely to points in $M$. Furthermore we prove that if $G$ is finitely generated and the random walk has finite entropy and finite logarithmic moment with respect to the word metric, then $M$, with the corresponding hitting measure, can be seen as a model for the Poisson boundary of $G$.