There are many families of Calabi-Yau varieties with huge groups of birational automorphisms. The following example we learned from D.Panov and D.Zvonkine. For any real numbers $l_{1},\dots,l_{n}>0$ we can consider the space of planar $n$-gons with the length of edges equal to $l_{1},\dots,l_{n}$, modulo the group of orientation-preserving motions. This space can be identified with the space of solutions of the following system of equations

$$\sum l_{i}z_{i}=0,\,\,\,\sum l_{i}z_{i}^{-1}=0$$

where $(z_{1}:\dots:z_{n})\in{\bf C}P^{n-1}$ is a point satisfying the reality condition $|z_{i}|=1,\,\,i=1,\dots,n$. Hence we obtain a singular subvariety of ${\bf C}P^{n-1}$ of codimension $2$, depending on parameters $l_{1},\dots,l_{n}$. One can check that this variety is birationally isomorphic to a non-singular Calabi-Yau variety. For any proper set $I\subset\{1,\dots,n\}$, $2\leq|I|\leq n-2$ we have a birational involution $\sigma_{I}$ defined by the formula

$$\sigma_{I}^{*}(z_{i})=\left\{\begin{array}[]{ll}c/z_{i}&\mbox{ if $i\in I$}\\ z_{i}&\mbox{ if $i\notin I$}\end{array}\right.$$

where $c:=\frac{\sum_{i\in I}l_{i}z_{i}}{\sum_{i\in I}l_{i}/z_{i}}$.

We do not know at the moment the structure of the group $G_{n}$ generated by involutions $\sigma_{I}$. One can obtain easily explicit formulas for the action of $G_{n}$ by piecewise-linear homemorphisms of $S^{n-3}$. Length parameters $l_{i}$ should be replaced by elements of a non-archimedean field $K$ with “generic” norms $\lambda_{i}=val_{K}(l_{i})\in{\bf R}$. Denote by $\zeta_{i},\,\,i=1,\dots,n$ real variables which have the meaning of valuations of variables $z_{i}\in K$. Sphere $S^{n-3}$ is obtained in the following way. In ${\bf R}^{n}$ we consider the intersection of two subsets:

$$\{(\zeta_{1},\dots\zeta_{n})\,|\,\,\,\min_{i}(\lambda_{i}+\zeta_{i})\mbox{ is achieved at least twice}\}$$

and

$$\{(\zeta_{1},\dots\zeta_{n})\,|\,\,\,\min_{i}(\lambda_{i}-\zeta_{i})\mbox{ is achieved at least twice}\}$$

and then take the quotient by the action of ${\bf R}$:

$$(\zeta_{1},\dots\zeta_{n})\rightarrow(\zeta_{1}+c,\dots\zeta_{n}+c)$$

corresponding to the projectivization. For appropriately chosen $(\lambda_{1},\dots,\lambda_{n})$ we obtain a set which is the union of $S^{n-3}$ with several “wings” going to infinity. The action of the involution $\sigma_{I}$ is obtained from algebraic formulas from above, in which one replace non-archimedean variables by real ones, addition by minimum and multiplication (division) by addition (subtraction).