Genericity of Filling Elements

On the genericity of loxodromic actions

One way of picking a "generic" element of a finitely generated group is to pick a random element with uniform probability in a large ball centered on $1$ in the Cayley graph. If the group acts on a $δ$-hyperbolic space, with at least one element acting loxodromically, then it is plausible that generic elements should act loxodromically with high probability. In this paper we prove that the probability of acting loxodromically is bounded away from 0, provided the group satisfies a very weak automaticity condition, and provided a certain compatibility condition linking the automatic with the $δ$…

Genericity of contracting elements in groups

In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 elements in CAT(0) groups, and the set of pseudo-Anosov elements in mapping class groups. Regarding a proper action, the set of non-contracting elements is proven to be growth-negligible. In particular, for mapping class groups, the set of pseudo-Anosov elements is generic in a su…

群中压缩测地线的普遍性

Genericity of contracting geodesics in groups

本文通过Cayley图中压缩元素的普遍性，刻画了有限生成群的Gromov双曲性。

柱面双曲群中的增长间隙与指数一般性

Growth gaps and exponential genericity in acylindrically hyperbolic groups

研究柱面双曲群，证明对其每个有限生成集，非WPD元素集指数增长率更小，即WPD元素指数一般，还证明了该群的增长紧性和余增长紧性。

On the genericity of Whitehead minimality

We show that a finitely generated subgroup of a free group, chosen uniformly at random, is strictly Whitehead minimal with overwhelming probability. Whitehead minimality is one of the key elements of the solution of the orbit problem in free groups. The proofs strongly rely on combinatorial tools, notably those of analytic combinatorics. The result we prove actually depends implicitly on the choice of a distribution on finitely generated subgroups, and we establish it for the two distributions which appear in the literature on random subgroups.

Generation and genericity of the group of absolutely continuous homeomorphisms of the interval

We examine the Polish group $H_{+}^{AC}$ of order-preserving self-homeomorphisms $f$ of the interval $\left[ 0,1 \right]$ for which both $f$ and $f^{-1}$ are absolutely continuous; in particular, we establish two results. First, we prove that $H_{+}^{AC}$ is topologically $2$-generated; in fact, it is generically $2$-generated, i.e., there is a dense $G_δ$ set of pairs $\left( f,g \right) \in H_{+}^{AC} \times H_{+}^{AC}$ for which $\left\langle f,g \right\rangle$ is dense. Secondly, we prove that $H_{+}^{AC}$ admits a dense $G_δ$ conjugacy class, and we explicitly characterize the elements th…