TITLE: Roots in Thompson's groups
QUESTION [7 upvotes]: Let $G \in \{F,T,V\}$, and let $g$ be an element of Thompson's $G$.

Question 1. When does $g$ have an $n$th root?

If this is a complicated issue, and a description is not so easy to give, I am especially interested in the following more specific questions (for any of the $G$).

Question 2. Given $(g,n)$, is it decidable whether $g$ has an $n$th root?


Question 3. Can an element $g$ have roots of arbitrarily large orders?

For $G = F$, if $g \neq 1_F$, looking at the slope at the leftmost point of $g$'s support we get an upper bound on the possible $n$'s, so Question 3 has a negative answer for this one. For $T$ and $V$ I don't see any trivial argument. I feel like Question 2 should have an easy "yes" answer, but I don't know how to prove that.

REPLY [2 votes]: More or less to add to Jim's answer: The strand diagram material is equivalent to Brin's Revealing Pair technology (introduced in his paper Brin's Higher Dimensional Thompson groups where he introduced $2V$).  We use this revealing pair technology in fashion very similar to Jim's description for $F$ elements above to describe roots of elements in $V$ in (mostly Section 7) of our paper "Centralizers in the R. Thompson group $V_n$" published in GGD in 2013 (see Centralizer's in $V$).
The main point is that the centralizer of a non-torsion element is virtually $Z$ over the region of the support of the element in Cantor space.  So, there is a finite set of "rootiest roots" supported there.  If our element $g$ has root $r$ in this set, and $r^k=g$, then away from the support of $r$, one could put an infinitude of torsion elements of order dividing $k$.  All of these created elements are also "roots" of $g$, but in a stupid way.
Ewa Bieniecka in her thesis developed and generalized the revealing pair technology further, and answered the other direction of the results characterized in our centralizer's paper.  I think this is a very gentle resource for learning flow graphs and revealing pairs (the crucial technologies).