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 $\alpha_A$

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 $\alpha_A$. These “one-quiver settings” are easy to

work out for a group algebra $lG$ if $G$ is the amalgamated free product

of finite groups $G = H_1 \bigstar_H H_2$.

Here is how to do

this : construct a bipartite quiver with the left vertices corresponding

to the irreducible representations of $H_1$, say ${ S_1, .. ,S_k }$ of

dimensions $(d_1, .. ,d_k)$ and the right vertices corresponding to the

irreducible representations of $H_2$, ${ T_1, .. ,T_l }$ of dimensions

$(e_1, .. ,e_l)$. The number of arrows from the $i$-th left vertex to

the $j$-th right vertex is given by the dimension of $Hom_H(S_i,T_j)$

This is the quiver I call the Zariski quiver for $G$ as the finite

dimensional $G$-representations correspond to $\theta$-semistable

representations of this quiver for the stability structure $\theta=(d_1,

.. ,d_k ; -e_1, .. ,-e_l)$. The one-quiver $Q(G)$ has vertices

corresponding to the minimal $\theta$-stable dimension vectors (say

$\alpha,\beta, .. $of the Zariski quiver and with the number of arrows

between two such vertices determined by $\delta_{\alpha

\beta}-\chi(\alpha,\beta)$ where $\chi$ is the Euler form of the Zariski

quiver. In the old note I've included the example of the projective

modular group $PSL_2(Z) = Z_2 \bigstar Z_3$ (which can easily be

generalized to the modular group $SL_2(Z) = Z_4 \bigstar_{Z_2} Z_6$)

which turns out to be the double of the extended Dynkin quiver

$\tilde{A_5}$. In the rewrite I've also included an example of a

congruence subgroup $\Gamma_0(2) = Z_4 \bigstar_{Z_2}^{HNN}$ 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 $GL_2(Z) = D_8 \bigstar_{Z_2

\times Z_2} (S_3 \times Z_2)$where $D_8$ 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.

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