As the title says, I'm interested in finding out what is meant by Cohomology Tree Probability Distribution. I came across this term on group props, and it seems to be a distribution for randomly selecting a group of a given order.
At first, it seems like this is a simple Google-able request, but surprisingly, searching for "Cohomology Tree Probability Distribution" (in quotes) yields only hits from group props. Moreover, group props itself doesn't even have a page on it!
It only seems to be listed for various $p$-groups, which makes me suspect that it has something to do with nilpotency, but I really have no idea. Has anyone heard of this? Does anyone know the definition? Can someone point me towards a resource to read about this?
I didn't find any resource to read about this, but I think I can give you a definition:
The basic idea behind this is the notion of a central extension:
Definition A central extension of $G$ by an abelian group $A$ is a short exact sequence $$ \mathbb E \colon \quad 1 \rightarrow A \rightarrow E \rightarrow G \rightarrow 1$$ such that the image of $A$ is in the center of E. An isomorphism between central extensions $\mathbb E$ and $\mathbb E'$ is a group isomorphism $\alpha \colon E \rightarrow E'$ such that $$\require{AMScd} \begin{CD} 1 @>>> A @>>> E @>>> G @>>> 1\\ @. @| @V{\alpha}VV @| @. \\ 1 @>>> A @>>> E' @>>> G @>>> 1\\ \end{CD} $$ commutes. The set of isomorphism classes of central extensions is one interpretation of the second cohomology group $H^2(G,A)$. This group is a finite abelian group if $G$ and $A$ are finite. Given a cohomology class $\eta$ in $H^2(G,A)$, we can consider an associated short exact sequence $$ \mathbb E(\eta) \colon \quad 1 \rightarrow A \rightarrow E \rightarrow G \rightarrow 1$$ and look at the isomorphism class of $E$. If $A$ and $G$ are finite, we obtain a map from $H^2(G,A)$ to the set of isomorphism classes of groups of order $|A| \cdot |G| $. This map is rarely injective. For example, all non-zero elements $H^2(\mathbb F_p, \mathbb F_p) \cong \mathbb F_p$ are sent to $\mathbb Z / p^2\mathbb Z$.
Now we define a stochastic process $(G_n)_{n\in \mathbb N}$ where $G_n$ will be a group (or rather an isomorphism class of groups) of order $p^n$. We start with $G_1 = \mathbb F_p$ the cyclic group of order p. If we have already constructed $G_n$, we take a uniformly random element of $\eta \in H^2(G_n,\mathbb F_p)$ and set $G_{n+1}$ to the middle term of $\mathbb E(\eta)$.
If $G$ is any group order $p^n$, then we can show by induction that $\mathbb P(G_n \cong G) >0$: This is obviously true for $n=1$, so let $n \gt 1$. Since $G$ is a p-group, the center $Z(G)$ is a non-trivial abelian p-group. If $A \subset Z(G)$ is a subgroup of order p, then
$$ 1 \rightarrow A \rightarrow G \rightarrow G/A \rightarrow 1$$ is a central extension. So by induction, $\mathbb P(G_n \cong G) \ge \mathbb P(G_n \cong G \mid G_{n-1} \cong G/A) \cdot \mathbb P(G_{n-1} \cong G/A) > 0$.
You can probably implement this process using GAP. I found some references on group props and the GAP manual.