TITLE: Groupoid of moves on trivalent fatgraph
QUESTION [13 upvotes]: $\DeclareMathOperator\Out{Out}$Let $T$ be a finite trivalent fatgraph — i.e. a graph with a cyclic order of the edges at each vertex. Then there are certain basic "moves" we can perform on $T$: an embedded edge can be collapsed and then uncollapsed in a different way (a "rotation", or "2-2 move"), or the circular order of the three edges incident at a vertex can be reversed (a "flip").
Define a set $\mathcal{T}$ whose elements are trivalent fatgraphs $T'$ homotopic to $T$ with a labeling of the edges from $1$ to $n$ and a labeling of the vertices from $1$ to $m$ (note $m = 2n/3$). A "move" is a pair $(T,c)$ where $T \in \mathcal{T}$, and $c$ is an element of the set $e_1, e_2, \cdots, e_n, v_1, v_2, \cdots, v_m$. The move acts on the labeled fatgraph $T$, and turns it into a new labeled fatgraph $T'$ obtained from $T$ by performing a rotation on edge $e_i$ if $c=e_i$ or a flip on vertex $v_i$ if $c=v_i$. It is clear how a flip affects the labels (it doesn't). A rotation destroys one edge labeled by $e_i$ and creates a new edge, so label this new edge $e_i$.
Now define a marked fatgraph to be a labeled fatgraph (i.e. an element of $\mathcal{T}$) together with a homotopy class of homotopy equivalence to some fixed $K(\pi_1(T),1)$. The moves defined above generate a new groupoid on marked fatgraphs, by acting on the labeled fatgraph part. This groupoid — the groupoid acting on marked fatgraphs — I will denote by $V(T)$ (the notation $V(T)$ is suggested by the similarity to Thompson's group $V$).
This groupoid — or something like it — turns up in many different contexts, so as a preliminary question, it would be nice to know how it is referred to. (Or: does this construction even make sense?)
More substantially: what is known about the algebraic structure of $V(T)$? What can be said about the cohomology of its classifying space? What is the relation to the group $\Out(F)$, where $F$ is the (free) fundamental group of $T$? (note that $\Out(F)$ acts on marked fatgraphs in a way that commutes with $V(T)$, by acting by homotopy equivalences of the $K(\pi_1(T),1)$ and thereby changing the marking). Is there a good reference?

Since the answers I am getting are not really what I am after, I think I need to make the question more pointed. A marked fatgraph determines a certain amount of algebraic structure  on a free group (i.e. the fundamental group of $T$), namely a pair $(l,e)$ where $l$ is a length function, and $e$ is a bounded 2-cocycle. The first part of the data comes from the "thin" underlying graph, and is just the translation length of each element on its axis. The second part of the data comes from the fattening, and is an explicit cocycle representing the Euler class of the thickened surface. The first kind of move affects $l$, the second kind affects $e$. Crucially, both $l$ and $e$ are integer valued (this is the point of discussing discrete combinatorial objects, namely fatgraphs, instead of e.g. discrete faithful representations of $F$ into $\operatorname{PSL}(2,\mathbb{R}))$.
Many, many papers discuss length functions, and many, many papers discuss Euler classes, but I would like to have a (presumably homological) algebraic framework which treats the two components as a single object with, presumably, more structure. The question is: what is this structure? Is it something that is already well-studied? Is there a reference?

REPLY [2 votes]: This is a nice question and I find it very hard to make such things understandable so I'm impressed.
The relationship to Outer Space of course looks very strong, but you're just looking at the simplicies of maximal dimension (trivalent graphs), so to understand the relationship my first step would be to try to lift your moves to the whole of outer space.  
Your edge moves get factored into edge expansion and edge contraction, these are well understood and studied, see any reference on outer space.  As for the vertices this means having to alter fat vertices of higher degree and looking at the trivalent case I think that means that you have to allow moves between any two cyclic orderings.  A first glance seems to indicate that there is some kind of compatibility between edge and vertex moves: essentially it does look to me that you can lift your concept of moves to the whole of "fattened outer space".
Now we've done this we can study your object in the formalism of Kontsevich's graph complexes (well the associated (semi-)simplicial structures).  Fattened outer space becomes the associative graph complex but with some extra simplices (your vertex moves).  We can characterise these as certain simplices coming from the kernel of the map from the associative graph complex to the commutative graph complex.
As such I don't think that your groupoid will mean much beyond outer space, but it may encode some information from the associative complex, although I'm guessing only something along the lines of enumeration of the kernel.
I've run out of time and need to leave, hopefully what I've said makes some sense and the details are fill-in-able.  I'll address any comments tomorrow.