Modular forms (Lent 2011) — example sheet #2
... Γ ⊃ Γ(N ) where N is the least common multiple of the widths of the cusps of Γ. (This gives a way to tell whether or not a given group is a congruence subgroup.) (iii) Assuming (ii), write down an infinite family of subgroups of Γ0 (11) which are not congruence subgroups. (Recall that the compact Ri ...
... Γ ⊃ Γ(N ) where N is the least common multiple of the widths of the cusps of Γ. (This gives a way to tell whether or not a given group is a congruence subgroup.) (iii) Assuming (ii), write down an infinite family of subgroups of Γ0 (11) which are not congruence subgroups. (Recall that the compact Ri ...
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY 1
... elegant fact that the fundamental group of any linear graph is a free group. Definition 5.1. Let X be a space; let x0 be a point of X. The set of path homotopy classes of loops based at x0 , with the operation *, is called the fundamental group of X relative to the base point x0 . This fundamental g ...
... elegant fact that the fundamental group of any linear graph is a free group. Definition 5.1. Let X be a space; let x0 be a point of X. The set of path homotopy classes of loops based at x0 , with the operation *, is called the fundamental group of X relative to the base point x0 . This fundamental g ...
The Lebesgue Number
... cover of the compact metric space X. By the above lemma, we only have to let λ be a Legesgue number for V. For if now S is any subset of X of diameter less than λ, then S ⊆ f −1 (U ) for some U ∈ U, which means f (S) ⊆ U . Corollary. Suppose f : X → Y is a continuous function from a compact metric s ...
... cover of the compact metric space X. By the above lemma, we only have to let λ be a Legesgue number for V. For if now S is any subset of X of diameter less than λ, then S ⊆ f −1 (U ) for some U ∈ U, which means f (S) ⊆ U . Corollary. Suppose f : X → Y is a continuous function from a compact metric s ...
