On genericity of pseudo-Anosovs in the Torelli group

Genericity of pseudo-Anosov mapping classes, when seen as mapping classes

We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.

On the genericity of pseudo-Anosov braids II: conjugations to rigid braids

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated "easily" into a rigid braid.

On the genericity of pseudo-Anosov braids I: rigid braids

We prove that, in the $l$-ball of the Cayley graph of the braid group with $n \geqslant 3$ strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of $l$ by a positive value.

闭曲面Torelli群中伪Anosov映射的迹数域次数

Trace field degrees in the Torelli group

研究证明闭曲面Torelli群中伪Anosov映射的迹数域次数范围，并通过实例验证了Thurston的断言。

从注入空间出发的词度量中的增长紧密性与普遍性

Growth tightness and genericity for word metrics from injective spaces

该研究通过几何方法证明，从注入空间中获取的有限生成集对应的词度量中，伪Anosov映射呈指数普遍性，并验证了Cayley图的增长紧密性。

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…