Dead ends on wreath products and lamplighter groups

Abstract

For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=\mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

Figures (6)

• Figure 1: Paths visiting all vertices in the square $[-n,n]^2$, starting at $(0,0)$ and finishing inside the square.
• Figure 2: The Cayley graph $\mathrm{Cay}(\mathbb{Z},\{\pm 1,\pm 2\})$.
• Figure 3: There are no Hamiltonian paths from $u$ to $v$ in these grid graphs.
• Figure 4: Inductive step of the proof of Lemma \ref{['lem: rectangles are quasi-Hamiltonian-connected']}.
• Figure 5: Each octagon defines a "partition by petals" of $\mathrm{Cay}(\mathbb{Z}/8\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}, \{b,c\})$.

Theorems & Definitions (49)

• Theorem A: = Theorem \ref{['thm: existence of genset with deep dead ends in any lamplighter']}
• Theorem B: = Theorem \ref{['thm: general dead ends on free products']}
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Proposition 2.5: ChenQuimpo1981
• Definition 2.6
• Lemma 2.7: Fuzz Lemma
• Definition 2.8