Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Biggs tree groups

Christopher H. Cashen

TL;DR

The paper rigorously identifies the isomorphism type of Biggs tree groups $G_{C,R}$ generated from a colored tree, showing that for $C\ge 3$ these groups are either $\mathrm{Alt}(N_{C,R})$ (when $C$ and $R$ are even) or $\mathrm{Sym}(N_{C,R})$ (otherwise), with an explicit generating set of size $C$. The authors establish that $G_{C,R}$ acts 2-transitively on the vertex set via a track-based action on $T_{C,R}$, and they apply Jones's theorem to limit possibilities to alternating or projective semilinear groups. By ruling out projective semilinear cases through cycle structures and order bounds, and using an inductive argument across the number of colors, they achieve a complete classification for $C>2$, together with constructive, explicit generators and a girth bound that grows with radius $R$. This work yields explicit, small-generator Cayley graphs of symmetric groups with unbounded girth, linking an accessible combinatorial construction to deep permutation-group classification with potential implications for graph properties beyond expansion alone.

Abstract

Biggs gave an explicit construction, using finite colored trees, of finite permutation groups whose Cayley graphs have valence \(C\) and girth tending to infinity as the radius \(R\) of the tree tends to infinity. We show that when the number of colors is at least 3, the group so presented contains the full alternating group on the vertices of the tree. This gives, for each \(C\geq 3\), an infinite family of pairs \((G_{C,R},S_{C,R})\) such that \(G_{C,R}\) is an alternating or symmetric group, \(S_{C,R}\) is a generating set of \(G_{C,R}\) of size \(C\) with an explicit permutation description of its generators, and such that the sequence of Cayley graphs \(\mathrm{Cay}(G_{C,R},S_{C,R})\) has constant valence \(C\) and girth tending to infinity as \(R\) tends to infinity.

Biggs tree groups

TL;DR

The paper rigorously identifies the isomorphism type of Biggs tree groups generated from a colored tree, showing that for these groups are either (when and are even) or (otherwise), with an explicit generating set of size . The authors establish that acts 2-transitively on the vertex set via a track-based action on , and they apply Jones's theorem to limit possibilities to alternating or projective semilinear groups. By ruling out projective semilinear cases through cycle structures and order bounds, and using an inductive argument across the number of colors, they achieve a complete classification for , together with constructive, explicit generators and a girth bound that grows with radius . This work yields explicit, small-generator Cayley graphs of symmetric groups with unbounded girth, linking an accessible combinatorial construction to deep permutation-group classification with potential implications for graph properties beyond expansion alone.

Abstract

Biggs gave an explicit construction, using finite colored trees, of finite permutation groups whose Cayley graphs have valence and girth tending to infinity as the radius of the tree tends to infinity. We show that when the number of colors is at least 3, the group so presented contains the full alternating group on the vertices of the tree. This gives, for each , an infinite family of pairs \((G_{C,R},S_{C,R})\) such that is an alternating or symmetric group, is a generating set of of size with an explicit permutation description of its generators, and such that the sequence of Cayley graphs \(\mathrm{Cay}(G_{C,R},S_{C,R})\) has constant valence and girth tending to infinity as tends to infinity.

Paper Structure

This paper contains 7 sections, 14 theorems, 16 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Comparison to other results
  3. Primitive permutation groups containing a cycle
  4. Tracking the action
  5. 2--transitivity
  6. Proof of the main theorem
  7. Further remarks

Key Result

Theorem 2.1

If $G$ is a primitive permutation group of degree $n$ containing a nontrivial cycle $g$ then either $G$ contains $\mathrm{Alt}(n)$ or $g$ fixes $0\leq k\leq 2$ points and one of the following is true:

Figures (7)

  • Figure 1: $\mathop{\mathrm{Cay}}\nolimits(G_{\{r,g,b\},1},\{r,g,b\})$, vertices labeled by permutations of vertices of $T_{3,1}$ with $o=0$, $o.r=1$, $o.g=2$, and $o.b=3$.
  • Figure 2: $T_{3,2}$ with the track of $(rgb)^{10}$ at $o$, showing that $rgb$ is a 10--cycle. The vertex is numbered $n$ when that vertex is $o.(rgb)^n$.
  • Figure 3: In $G_{3,7}$ suppose $v$ is at height 7 on the $(r,b)$--dichrome arc containing $o$. All alternating dichrome words of length 15 fix $o$, but none of them move $v$, $v'$, or $v"$ closer to $o$, since the dichrome arcs on which these vertices sit have lengths dividing 15. However, an $(r,g)$ alternating word of length 15 swaps $v$ and $v'$, a $(b,g)$ alternating word of length 15 swaps $v'$ and $v"$, and an $(r,b)$ alternating word of length 15 swaps $v"$ and $v"'$. Finally, the $(r,g)$--dichrome arc containing $v"'$ has 7 vertices, prime relative to 15. The alternating word $rg\cdots r$ of length 15 moves $v"'$ closer to $o$.
  • Figure 4: The lower bound for $\operatorname{log}(\mathop{\mathrm{meo}}\nolimits(G_{3,R}))$ is already greater than the upper bound for $\operatorname{log}(\mathop{\mathrm{meo}}\nolimits(\mathop{\mathrm{P\Gamma L}}\nolimits_d(q))$ when $R=6$.
  • Figure 5: The lower bound for $\operatorname{log}(\mathop{\mathrm{meo}}\nolimits(G_{4,R}))$ is already greater than the upper bound for $\operatorname{log}(\mathop{\mathrm{meo}}\nolimits(\mathop{\mathrm{P\Gamma L}}\nolimits_d(q))$ when $R\geq 3$.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Theorem 2.1: Excerpt of Jon14
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 16 more