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Tuesday, January 24, 2017

**Abstract:** For a given finite index subgroup $H\subseteq$SL(2,$\mathbb{Z}$), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to the finite cover SL(2,$\mathbb{R}$)/$H$ of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$). We then relate the properties of this section to the gaps in Farey fractions and describe how the ergodic properties of the horocycle flow can be used to obtain certain statistical properties of various subsets of Farey fractions.

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