We present the formal definition of Closing Steinberg symbols.
Let $\Gamma$ be a group acting on a space $X$ by group action $X \times \Gamma \to X$. If $P \subset X$ is a subset of $X$, then a finite subset $I$ of $\Gamma$ is said to formally close $P$ if the iterated symmetric difference of $\{ \gamma. P ~|~ \gamma\in I\}$ vanishes (equal to empty set $\emptyset$).
N.B. The vanishing of the iterated symmetric difference means the chain sum $\gamma.P$, for $\gamma \in I$, vanishes over mod 2 coefficients. I.e. every element in the $I$-translates of $P$ occurs an even number of times.
The above is the abstract formulation of a problem we call Closing the Steinberg symbol. In our applications the subset $P$ is a panel representing the convex hull of a sphere at infinity which is called the "Steinberg symbol".
Now the key to Closing Steinberg (CS) is to find nontrivial formal solutions. We will illustrate with the specific group $\Gamma = Mod(S)$ which is the mapping class group of a compact hyperbolic surface $S$. We will begin with genus $g(S)=2$.
Now we introduce the basic functions using Mark C. Bell's curver:
Now we make the basic definitions of the reference pant, and the mapping class elements $\zeta, \nu, \mu$.
pant={a,b,c} ## pant={a,b,c} is the standard pant.
zeta=a*e*c*f*b ## zeta is the order 6 element in MCG arising from chain relation. ##
nu=a*e*c*f ## nu is order 10 element in MCG
mu=nu**4 ## mu is order 5 element in MCG.
We define $\xi$ as the union of the standard pair of pants with its $\mu$-translate. The $\mu$-translate of $\{a,b,c\}$ is a pair of pants dual to $\{a,b,c\}$.
The mapping class element $\mu$ is an order 5 element in $Mod(S_2)$. We propose that the powers of $\mu$, namely $I=\{Id, \mu, \mu^2, \mu^3, \mu^4\}$, formally close the symbol $\xi$. This solution will be nontrivial because we will establish that $\mu^i\xi = \mu^j\xi$ if and only if $i=j$. Thus the $I$-translates of $\xi$ are distinct, while the chain sum $\sum_{i=0}^4 \mu^i\xi$ vanishes over ${\bf{Z}}/2$ coefficient.
## xi is obtained by joining the initial pant p with its mu translate.
xi=pant|Translate(mu, pant)
## important to verify that pant and the mu-translate are disjoint.
## Ad(mu,pant) is "opposite pair of pants"
print("The mu translate of the standard pant is disjoint from pant. ", pant & Translate(mu, pant) == set())
print()
M0=xi
M1=Translate(mu,xi)
M2=Translate(mu**2,xi)
M3=Translate(mu**3,xi)
M4=Translate(mu**4,xi)
## The following proves that all the symbol translates are nontrivial, and there is no complete coincidence
## between the translated symbols.
print("The mu translates of xi are all pairwise distinct:", M0!=M1 and M0!=M2 and M0!=M3 and M0!=M4 and M1!=M2 and M1!=M3 and M1!=M4 and M2!=M3 and M2!=M4 and M3!=M4 )
print()
## The following proves that the total chain sum of the translated symbols vanishes mod 2.
## I.e. the iterated symmetric difference of the translated symbols is equal to empty set.
print("The iterated symmetric difference of the mu translates is empty.", ((((M0^M1))^M2)^M3)^M4 ==set())
print()
print("The mu-orbit of xi is supported on ten curves.", 10==len(M0|M1|M2|M3|M4) )
print()
print("Therefore we find I={Id, mu, mu**2, mu**3, mu**4} is a formal solution to Closing the Steinberg symbol xi in genus two.")
print("")
So we have found a formal solution $I$ to CS. For applications we need further verify that $I$ satisfies further geometric properties. Specifically we need establish:
- the $I$-translates of $\xi$ have a well-defined convex hull $F:=conv(I.\xi)$ in $Teich(S)$.
- the $\Gamma$-translates of $F$ generate a chain sum $\underline{F}:=\sum_{\gamma \in \Gamma} \gamma.F$ with well separated gates equal to the $\Gamma$-translates of $\xi$.
Informally the idea is that $\xi$ represents a "panel" $P$, and $I$ closes the panel in the sense that the $I$-translates of the panel assemble to a closed ball. (Similar to how the (triangular, hexagonal) panels of a soccer ball assemble to form the closed ball).
But we must further study the $\Gamma$-translates of the ball itself, i.e. of the convex hull $F$. Most important for our setting is that the intersections of the various translates $\gamma F \cap \gamma' F$ have a "standard form", namely isometric to $\xi$ (the panel $P$).
Remark. It's not clear whether the above verification of ``well-separated gates" can be performed in curver. While we are capable of considering the $\Gamma$ action on the elements of $I.\xi$, we cannot necessarily compute the intersections $F\cap \gamma F$. The issue is that $F\cap \gamma F$ can intersect "at-infinity" (i.e. be asymptotic) without the convex hulls having an intersection in the interior of $Teich(S)$. Formally the solutions to CS solve a problem at infinity, but the next step is to study the solutions in the interior, and this becomes more geometric.