Editing Projective linear group (section)

=== Hurwitz surfaces ===
[[File:3-7 kisrhombille.svg|thumb|Some PSL groups arise as automorphism groups of Hurwitz surfaces, i.e., as quotients of the [[(2,3,7) triangle group]], which is the symmetries of the [[order-3 bisected heptagonal tiling]].]]
{{see|Hurwitz surface}}
PSL groups arise as [[Hurwitz group]]s (automorphism groups of [[Hurwitz surface]]s – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, the [[Klein quartic]] (genus 3), has automorphism group isomorphic to {{nowrap|PSL(2, 7)}} (equivalently {{nowrap|GL(3, 2)}}), while the Hurwitz surface of second-lowest genus, the [[Macbeath surface]] (genus 7), has automorphism group isomorphic to {{nowrap|PSL(2, 8)}}.

In fact, many but not all simple groups arise as Hurwitz groups (including the [[monster group]], though not all alternating groups or sporadic groups), though PSL is notable for including the smallest such groups.