Fricke groups and Fricke curves

Question

The congruence subgroup $\Gamma_{0}(n) \subset PSL_{2}(\mathbb{Z})$ is normalized by the Fricke involution $F_n: z \mapsto -1/nz$ and so we may form the Fricke modular group $\Gamma_{0}^{+}(n)\langle \Gamma_{0}(n), F_n \rangle \subset PSL_{2}(\mathbb{R})$. (Sometimes people focus on the case $p=n$, since then I think that $\Gamma_{0}^{+}(p)$ is the full normalizer of $\Gamma_{0}(p) \subset PSL_{2}(\mathbb{R})$.)

Questions:

1. Is it known that and how $\Gamma_{0}^{+}(n)$ splits as a free product of cyclic groups? (This is known for $\Gamma_{0}(n)$ and I can check it by hand for $n=2$.)

2. Are convenient fundamental domains for $\Gamma_{0}^{+}(n)$ acting on the upper half-plane $\mathbb{H}$ known?

3. Are the orbifold points of $\mathbb{H}/\Gamma_{0}^{+}(n)$ completely understood?

For questions of the form `Is it known or understood... ?', I'm looking for answers of the form 'Yes, this is well-known and works as follows... or is written down here...' or 'Yes, it is well-known, but scattered over the literature...' or 'No, this is not obvious.' Also, any good reference is welcome, since I've only found bits and pieces addressing these questions.

Answer 1

The answer to question 1. is yes. The quotient $\mathbb{H}^2/\Gamma_0^+(n)$ is an orbifold with cusps. Take a separating non-trivial arc connecting a cusp to itself. One may show that such an arc exists using the fact that the orbifold has negative Euler characteristic, and therefore has at least three cone points or cusps, or has positive genus (and at least one cone point). One may take an arc from a cusp to one cone point, and take the boundary of a regular neighborhood to get a separating arc which is non-trivial. By Van-Kampen's theorem, the fundamental group is a free product.

For question 2., check out Cummin's tables. Cummins has a list of fundamental domains for $\Gamma_0(n)$ for many such $n$ which he sent me once, and I think were used or are related to the computations of matrix generators. As Kevin Buzzard says, once you have the fundamental domain for $\Gamma_0(N)$, you can take half of it to get one for $\Gamma_0^+(N)$.

For question 3., I think this is explicitly known in terms of arithmetic data. See Johansson.

Answer 2

Following Agol's answer, you can find a list of genus zero groups of $n|h$-type together with a discussion of fundamental domain computations in Ferenbaugh's paper: The Genus-zero problem for n|h-type groups, Duke Math. J. 72, no. 1 (1993), 31-63 (sorry, subscription required). From that, you can find all integers $n$ for which the group $\Gamma_0(n)+n$ is genus zero. Here's the list of possible $n$:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59, 71.