Common overlattices in trees and trees with fins

TL;DR

This work investigates whether Bass–Kulkarni’s overlattice phenomenon for uniform lattices in $\operatorname{Aut}(T)$ extends to trees with fins. It develops generalized universal groups $U^{(l)}(F)$ via a $\tau$-legal edge labelling $l$ and a subgroup $F<S_n$, establishing a universal property that any vertex-transitive subgroup embeds into some $U^{(l)}(F)$ with local action $F$. The paper then constructs counterexamples in trees and trees with fins, showing pairs of free lattices with no common overlattice in $\operatorname{Aut}(\mathbf{X})$, thus evidencing a breakdown of the Bass–Kulkarni-type overlattice result beyond ordinary trees. In contrast, when labellings are legal, a lattice $\Lambda^{(l)}(F)$ exists containing a conjugate of every free uniform lattice in $U^{(l)}(F)$, recovering a version of the original theorem; this is transported to a tree with fins via a subdivision $T^*$ with $\operatorname{Aut}(\mathbf{T^*})\cong U^{(l)}(F)$. The results reveal a sharp dichotomy: nonexistence of common overlattices in general trees with fins, versus guaranteed overlattices in the legal universal-group setting, and they raise natural questions about when such overlattice phenomena occur.

Abstract

Bass and Kulkarni proved that any pair of free uniform lattices in the automorphism group of a tree have conjugates that both lie inside a third uniform lattice (which is not necessarily free). We show that this does not generalise to trees with fins. The construction of our counter-example involves working with a certain generalisation of the universal groups of Burger and Mozes.

Figures (3)

• Figure 1: The labelled graphs $X$ and $X'$ used in Example \ref{['exmp:nooverlattice']}. Note that $X'$ is obtained from $X$ by cyclically permuting the labels $181,182,\dots,240$. Note that each label is shown near the origin end of the edge, for example the edge $e$ in $X$ has label 1 while the inverse edge $\bar{e}$ has label 61.
• Figure 2: The $l$ and $l"$-labellings on the edges $e_{121},\bar{e}_{121},e_{181},e_{182}$, as in the proof of Theorem \ref{['thm:univ']} (note that the labellings on $e_{121}$ are shown near the origin of the edge -- in this case near the vertex $x$ -- while the labellings on $\bar{e}_{121}$ are shown near the terminus of $e_{121}$).
• Figure 3: The map $\gamma_{x,f}:Y_{x,f}\to T^*$. The subtree $e^*\subset T^*$ indicated is associated with $e\in E(x)$ such that $l(e)=f(i)$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Remark 3.3