The definition of Higman-Thompson group

Question

The Thompson group is the set of piecewise linear increasing homeomorphisms from the closed unit interval to itself that are differentiable except at finitely many dyadic rational numbers and such that on intervals of differentiability the derivatives are powers of 2. There is a concept of Higman-Thompson groups which are generalization of Thompson group. I would like to know the definition of Higman-Thompson groups. Thank you in advance!

Answer 1

A good reference is Brown's article Finiteness properties of groups (1987). Section 4 of the paper defines and describes carefully the groups $G_{n,r}$, now referred to as Higman-Thompson groups. Actually, the section provides a good general introduction to Thompson groups. This is also the first place where the groups $F_{n,r}$ and $T_{n,r}$ appear. On pages 54-55, there are some interesting historical remarks.

For an accessible description of these groups, I think that a good point of view is provided by quasi-automorphisms of trees. Given a tree $\mathcal{T}$, a quasi-automorphism is a bijection $\mathcal{T}^{(0)} \to \mathcal{T}^{(0)}$ that preserves adjacency and non-adjacency for all but finitely many pairs of vertices [1]. Now, $\mathrm{QAut}(\mathcal{T})$ contains a natural normal subgroup: the subgroup $\mathrm{FSym}(\mathcal{T})$ of finitely supported bijections $\mathcal{T}^{(0)} \to \mathcal{T}^{(0)}$. The group of almost-isometries $\mathrm{AIsom}(\mathcal{T})$ is the quotient $\mathrm{QAut}(\mathcal{T})/ \mathrm{FSym}(\mathcal{T})$ [2].

Equivalently, an almost-isometry can be described as a triple $(A,B,f)$, where $A,B$ are finite subtrees and where $f : \mathcal{T} \backslash A \to \mathcal{T} \backslash B$ is an isometry; two almost-isometries $(A,B,f)$ and $(C,D,g)$ being identified whenever $f$ and $g$ agree on some cofinite subset. In pratice, this means that you remove two finite subtrees from two copies of $\mathcal{T}$ and permute the connected components by isometries.

From such a picture, you can impose restrictions on the permutations of components and on the isometries between components [3]. If $\mathcal{T}$ is a rooted binary tree (thought of as drawn on the plane), we obtain

• Thompson group $F$ if permutations and isometries preserve the left-right order induced by the plane;
• Thompson group $T$ if the permutations preserve the cyclic order on the components and the isometries the left-right order on each component;
• Thompson group $V$ if there is no restriction on the permutations and the isometries preserve the left-right order on each component.

But it is possible to modify the tree $\mathcal{T}$ and to obtain quite similar groups. Let $\mathcal{T}_{n,r}$ denote the tree with one vertex of degree $r$ while all the other vertices have degree $n+1$. Then the same definitions as above respectively give Thompson groups $F_{n,r}$, $T_{n,r}$, and $V_{n,r}$ (sometimes also denoted by $G_{n,r}$, following Higman).

[1] For instance, if $\mathcal{T}$ is a union of $n$ infinite rays with a common origin, then $\mathrm{QAut}(\mathcal{T})= H_n \rtimes S_n$ where $H_n$ is the $n$th Houghton group.

[2] For instance, if $\mathcal{T}$ is a binary rooted tree, then $\mathrm{AIsom}(\mathcal{T})$ is the Neretin group.

[3] This is the starting point of the definition of the so-called Röver-Nekrashevych groups.

Answer 2

Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). In the notation $G^+_{m,r}$, $m$ defines the variety, $r$ is the rank of the free algebra, $^+$ denotes the derived subgroup. In the Thompson case the algebras have operations $(\mu,\lambda,\rho)$ where $\mu$ is binary, $\lambda,\rho$ are unary and $\mu(\lambda(x),\rho(x))=x, \lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$ (somewhat similar to Hopf algebras. Thus if $A$ is a Jónsson–Tarski algebra, then $\mu$ is a bijection between $A\times A$ and $A$. These are finitely presented simple groups. It turned out that many groups $G^+_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.

Answer 3

Maybe I should also toss in the version of the definition that's most analogous to how the original question was phrased: The Higman-Thompson group $V_{n,r}$ is the group of piecewise linear increasing right-continuous bijections from $[0,r)$ to itself that are differentiable except at finitely many points in $\mathbb{Z}[1/n]\cap [0,r)$ and such that on intervals of differentiability the derivatives are powers of $n$.

Answer 4

Let $X$ be a structure consisting of finitely many unary and binary relations (which can be thought as some kind of coloring of vertices and oriented edges). If $Y$ is a subset of $X$, it is endowed with such a structure (restrict relations to $Y$). A subset $Y$ is called cofinite if $X\smallsetminus Y$ is finite. A structure isomorphism between structures is a bijection which preserves all relations.

The near automorphism group of $X$ consists of those structure isomorphisms between two cofinite subsets of $X$, identifying two such isomorphisms if they coincide on some smaller pair of cofinite subsets.

Now consider the (labeled) Cayley graph of the free semigroup $A_n$ on $n$ generators. The set is thus $A_n=\langle s_1,\dots,s_n\rangle$ and there are $n$ binary relations $R_i=\{(x,xs_i):x\in A_n\}$.

The near automorphism group of $A_n$ is the Higman-Thompson group $G_{n,1}$.

More generally, if we consider the disjoint union $A_{n,m}$ of $m$ copies of $A_n$, then its near automorphism group is $G_{n,m}$.

Note that removing the root in $A_n$ yields the disjoint union of $n$ copies of $A_n$. It follows that $A_{n,m}$ and $A_{n,m'}$ are near isomorphic if $n-1$ divides $m-m'$ (for $n\ge 2$ this is an iff). It follows that $G_{n,m}$ and $G_{n,m'}$ are near isomorphic if $n-1$ divides $m-m'$ (this is not an iff, Higman obtained partial results and Mark Sapir provides the reference to the final result).