On the intersection spectrum of $\operatorname{PSL}_2(q)$

Given a group $G$ and a subgroup $H \leq G$, a set $\mathcal{F}\subset G$ is called $H$\emph{-intersecting} if for any $g,g' \in \mathcal{F}$, there exists $xH \in G/H$ such that $gxH=g'xH$. The \emph{intersection density} of the action of $G$ on $G/H$ by (left) multiplication is the rational number $\rho(G,H)$, equal to the maximum ratio $\frac{|\mathcal{F}|}{|H|}$, where $\mathcal{F} \subset G$ runs through all $H$-intersecting sets of $G$. The \emph{intersection spectrum} of the group $G$ is then defined to be the set $$\sigma(G) := \left\{ \rho(G,H) : H\leq G \right\}.$$ It was shown by Bardestani and Mallahi-Karai [{\it J. Algebraic Combin.}, 42(1):111-128, 2015] that if $\sigma(G) = \{1\}$, then $G$ is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than $1$ belong to $\sigma(G)$, whenever $G$ is non-solvable. In this paper, we study the intersection spectrum of the linear group $\operatorname{PSL}_2(q)$. It is shown that $2 \in \sigma\left(\operatorname{PSL}_2(q)\right)$, for any prime power $q\equiv 3 \pmod 4$. Moreover, when $q\equiv 1 \pmod 4$, it is proved that $\rho(\operatorname{PSL}_2(q),H)=1$, for any odd index subgroup $H$ (containing $\mathbb{F}_q$) of the Borel subgroup (isomorphic to $\mathbb{F}_q\rtimes \mathbb{Z}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.