Year: 2025
Type: article
Abstract: Let X be an orientable hyperbolic surface of genus $$g\ge 2$$ g ≥ 2 with a marked point o, and let $$\Gamma $$ Γ be an orientable hyperbolic surface group isomorphic to $$\pi _{1}(X,o)$$ π 1 ( X , o ) . Consider the space $$\text {Hom}(\Gamma ,S_{n})$$ Hom ( Γ , S n ) which corresponds to n-sheeted covers of X with labeled fiber. Given $$\gamma \in \Gamma $$ γ ∈ Γ and a uniformly random $$\phi \in \text {Hom}(\Gamma ,S_{n})$$ ϕ ∈ Hom ( Γ , S n ) , what is the expected number of fixed points of $$\phi (\gamma )$$ ϕ ( γ ) ? Formally, let $$F_{n}(\gamma )$$ F n ( γ ) denote the number of fixed points of $$\phi (\gamma )$$ ϕ ( γ ) for a uniformly random $$\phi \in \text {Hom}(\Gamma ,S_{n})$$ ϕ ∈ Hom ( Γ , S n ) . We think of $$F_{n}(\gamma )$$ F n ( γ ) as a random variable on the space $$\text {Hom}(\Gamma ,S_{n})$$ Hom ( Γ , S n ) . We show that an arbitrary fixed number of products of the variables $$F_{n}(\gamma )$$ F n ( γ ) are asymptotically independent as $$n\rightarrow \infty $$ n → ∞ when there are no obvious obstructions. We also determine the limiting distribution of such products. Additionally, we examine short cycle statistics in random permutations of the form $$\phi (\gamma )$$ ϕ ( γ ) for a uniformly random
Source: Geometriae Dedicata
Author: Yotam Maoz
Institution: Tel Aviv University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,090
Award: G4544518169
OpenAlex ID: W4409519840