Intersection of subgroups in a surface group

Abstract: Let $G$ be a surface group, i.e the fundamental group of a compact surface. Denote by $d(G)$ the number of generators of $G$ and by $\chi(G)$ the Euler characteristic of $G$. We put $\bar \chi(G) = \max\{0, −\chi(G)\}$.

In this talk I will explain the following two results. In the first result we prove that for any two finitely generated subgroups $U$ and $W$ of $G$,

$$ \sum_{x \in U\backslash G / W} \bar \chi (U \cap x W x^{-1}) \le \bar \chi(U) \cdot \bar\chi(W). $$ From this we obtain the Strengthened Hanna Neumann conjecture for non-solvable surface groups. In the second result we show that if $R$ is a retract of $G$, then for any finitely generated subgroup $H$ of $G$, $$ d(R \cap H) \le d(H). $$ This implies the Dicks-Ventura inertia conjecture for free groups. The talk is based on a joint work with Yago Antolín.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.