Timeline for Coverings of a graph of groups
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 10, 2011 at 5:58 | vote | accept | joseph | ||
| Sep 5, 2011 at 20:25 | comment | added | HJRW | Here is a combinatorial description of an arbitrary covering space of the graph of groups that you're interested in. It has two classes of vertices, $u_i$ and $v_j$. Each $u_i$ is labelled by some $\mathbb{Z}/l_i$, where $l_i$ divides $l$, and each $v_j$ is labelled by $\mathbb{Z}/m_j$, where $m_j$ divides $m$; the valence of $u_i$ is $l/l_i$, and the valence of $v_j$ is $m/m_j$; finally, each edge adjoins one $u_i$ and one $v_j$. Does that make sense? | |
| Sep 5, 2011 at 20:20 | comment | added | HJRW | joseph - in answer to your first question, yes, the Bass--Serre tree does surject every connected cover. The trouble with defining it via the universal property is that such definitions contain almost no information. On the other hand, the fact that it is a tree is very useful - connectedness corresponds to the existence of normal forms, and simply-connectedness corresponds to uniqueness of normal forms. | |
| Sep 5, 2011 at 10:28 | comment | added | joseph | Also the Bass-Serre tree is just a tree; why we do not construct universal cover of graph of groups as again certain graph of groups with obvious universal property? | |
| Sep 5, 2011 at 10:28 | comment | added | joseph | @HW: The universal cover (Bass-Serre tree) of graph of groups is a tree on which the fundamental group of graph of group acts, with quotient graph of groups isomorphic to given graph of groups. But can we define it as a cover of given graph of groups which is also cover of any other cover of given graph of groups (similar to "topological universal cover")? I want to get all covers of graph of groups $\circ --\circ$, with vertex groups finite cyclic, edge group trivial. | |
| Sep 5, 2011 at 10:23 | history | edited | HJRW | CC BY-SA 3.0 |
Added reference to Scott and Wall.
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| Sep 5, 2011 at 10:16 | history | answered | HJRW | CC BY-SA 3.0 |