Measuring complexity of curves on surfaces

Coloring Curves on Surfaces

“…As a corollary of the lower bounds on self-intersection in Theorem 1.2, we prove that for "random" curves on a closed surface, the linear upper bounds on simple lifting degree from Arenas-Coto-Neumann [2] or from Aougab-Gaster-Patel-Sapir [1] in terms of intersection number can be improved to a square root upper bound: Corollary 1.6. When S is closed, there is some J ≥ 1 so that…”

“…One can still use π to pass between the ball of radius n in the punctured and closed surface groups, but a priori the fibers can have very different sizes, and perhaps a very large fiber sits over one of the very few curves in the closed surface with small self intersection. For this reason, we derive the P g statement of Theorem 1.2 from Theorem 1.3, using the work of Arenas-Coto-Neumann [2] as an intermediary (alternatively, one could also use the main theorem of [1]).…”

“…For example, Patel shows that deg(γ) can be bounded above by a linear function of the length of a geodesic representative for γ on a particular hyperbolic metric [21]. Aougab-Gaster-Patel-Sapir show that deg(γ) is bounded above by some linear function of the self-intersection number [1] (with constants depending on the complexity of the underlying surface), and Arenas-Coto-Neumann improve this to the uniform estimate deg(γ) ≤ 5i(γ, γ) + 5 [2,Thm. 1.1].…”

“…Simple lifting degree has been studied extensively, see for instance [21], [1], [9], [2]. For example, Patel shows that deg(γ) can be bounded above by a linear function of the length of a geodesic representative for γ on a particular hyperbolic metric [21].…”