15.12 (2002)

Let $G$ be a group acting faithfully and level-transitively by automorphisms on a rooted tree $\mathcal{T}$. For a vertex $v$ of $\mathcal{T}$, the rigid vertex stabilizer at $v$ consists of those elements of $G$ whose support in $\mathcal{T}$ lies entirely in the subtree $\mathcal{T}_v$ rooted at $v$. For a non-negative integer $n$, the $n$-th rigid level stabilizer is the subgroup of $G$ generated by all rigid vertex stabilizers corresponding to the vertices at the level $n$ of the tree $\mathcal{T}$. The group $G$ is a branch group if all rigid level stabilizers have finite index in $G$. For motivation, examples and known results see (R. I. Grigorchuk, in: New horizons in pro-$p$ groups, Birkhäuser, Boston, 2000, 121–179).

Do there exist branch groups with Kazhdan’s $\text{T}$-property? (See 14.34.)