*This semester the Hyperbolic Geometry Seminar, with the exception of a few dates, will not meet regularly*
February 5th: Robert Suzzi Valli (CUNY Graduate Center)
Title: Figure eights on triangle group orbifolds
Abstract: Let T be a triangle in the hyperbolic plane with angles π/p, π/q, π/r. The group Γ generated by the reflections in the three geodesic sides of T is a discrete group. The index two subgroup of the orientation preserving isometries of Γ is called a (p, q, r)-triangle group, denoted by Γ(p, q, r). If we consider the quotient of the hyperbolic plane by Γ(p, q, r) then the result is a triangle group orbifold, denoted O(p, q, r), which contains cone points of orders p, q, and r. Let C be a closed curve on O(p, q, r) disjoint from the cone points, having one self-intersection and whose two component loops bound distinct cone points (i.e., C is a figure eight curve). One might expect that the unique geodesic g in the free homotopy class of C is a figure eight curve. However, this is not always the case, and in this talk we determine all possibilities for g.
May 7th: Robert Suzzi Valli (CUNY Graduate Center)
Title: Paths and homotopy on orbifold surfaces
Abstract: Let
be the hyperbolic plane and Γ be a Fuchsian group, i.e., a discrete subgroup of the group of orientation preserving isometries of . If Γ contains elliptic elements then the quotient X =
/Γ is an orbifold surface containing cone points (the projection of elliptic fixed points). The presence of cone points on X require an adjusted notion of paths and homotopy on X, called X-homotopy, with the goal of defining the orbifold fundamental group as the set of X-homotopy classes of loops on X under the operation of concatenation. We will discuss this development along with the resulting important fact that the orbifold fundamental group is isomorphic to the Fuchsian group Γ (once base points are chosen).
Diversity and Inclusion at the GC