Before the vacation I finished a rewrite of the One quiver to rule them
all note. The main point of that note was to associate to any qurve
A (formerly known as a quasi-free algebra in the terminology of
Cuntz-Quillen or a formally smooth algebra in the terminology of
Kontsevich-Rosenberg) a quiver Q(A) and a dimension vector αA
such that A is etale isomorphic (in a yet to be defined
non-commutative etale toplogy) to a ring Morita equivalent to the path
algebra lQ(A) where the Morita setting is determined by the dimension
vector αA. These “one-quiver settings” are easy to
work out for a group algebra lG if G is the amalgamated free product
of finite groups G=H1★HH2.
Here is how to do
this : construct a bipartite quiver with the left vertices corresponding
to the irreducible representations of H1, say S1,..,Sk of
dimensions (d1,..,dk) and the right vertices corresponding to the
irreducible representations of H2, T1,..,Tl of dimensions
(e1,..,el). The number of arrows from the i-th left vertex to
the j-th right vertex is given by the dimension of HomH(Si,Tj)
This is the quiver I call the Zariski quiver for G as the finite
dimensional G-representations correspond to θ-semistable
representations of this quiver for the stability structure θ=(d1,..,dk;−e1,..,−el). The one-quiver Q(G) has vertices
corresponding to the minimal θ-stable dimension vectors (say
α,β,..of the Zariski quiver and with the number of arrows
between two such vertices determined by δαβ−χ(α,β) where χ is the Euler form of the Zariski
quiver. In the old note I've included the example of the projective
modular group PSL2(Z)=Z2★Z3 (which can easily be
generalized to the modular group SL2(Z)=Z4★Z2Z6)
which turns out to be the double of the extended Dynkin quiver
~A5. In the rewrite I've also included an example of a
congruence subgroup Γ0(2)=Z4★HNNZ2 which is an
HNN-extension. These are somehow the classical examples of interesting
amalgamated (HNN) groups and one would like to have plenty of other
interesting examples. Yesterday I read a paper by Karen Vogtmann called
Automorphisms of free groups and outer space in which I encountered
an amalgamated product decomposition for GL2(Z)=D8★Z2×Z2(S3×Z2)where D8 is the diheder group of 8
elements. When I got back from vacation I found a reference to this
result in my mail-box from Warren Dicks. Theorem 23.1, p. 82, in Heiner
Zieschang, Finite Groups of Mapping Classes of Surfaces, LNM 875,
Springer, Berlin, 1981.
I worked out the one-quiver and it has
the somewhat strange form depicted above. It is perfectly possible that
I made mistakes so if you find another result, please let me know.
added material (febr 2007) : mistakes were made and
the correct one quiver can be found elsewhere on this blog.