A *Gassmann-Sunada triple* is a triple $(U,V,W)$ of groups, with $V, W$ subgroups of $U$, such that $U$ and $V$ meet every conjugacy class in $U$ in the same number of elements, and such that $V$ and $W$ are not conjugate. Such triples have been widely used for constructing isospectral manifolds. However, usually when I encounter these triples in the manifold context, I read something like "... and if one now can find a suitable manifold such that $U$ acts in the right way, then ...."

My first question is:

(a) Starting from a triple $(U,V,W)$ as above, how does one naturally construct isospectral manifolds, which are not isometric ?

(So I want to see the manifolds arise only from the data $(U,V,W)$, and I also want a guarantee that the manifolds are not isometric.)

Next, I wonder if someone can give me a reference on the second question:

(b) Construct isospectral manifolds $\mathcal{M}_U$ and $\mathcal{M}_V$ from $(U,V,W)$ in the "natural way" (see the answers on (a)). Now let $T$ be a group properly containing $U$, and assume it "induces the same Gassmann-Sunada triple" (for instance, $T$ also acts on the same sets $X$ and $Y$ which give the permutation groups $(U,U/V)$ and $(U,U/W)$). My guess is that this "new" triple gives the same manifolds as those coming from $(U,V,W)$. Can anyone fill in the details here ?

Bye ...

surjection$\pi_1(M) \to G$. Thus, it wouldn't make sense to change $G$ to a larger $G'$. $\endgroup$ – David E Speyer May 11 '17 at 17:17