---
title: Small Coxeter Groups
url: https://www.emergentmind.com/topics/small-coxeter-groups
type: topic
---

# Small Coxeter Groups

Searching arXiv for the cited papers to ground the article in current literature.
Small Coxeter groups occur in two precise senses in recent work. In the rank-theoretic framework of the Coxeter galaxy, the term refers to the first three layers \(G_{\le 3}\), namely Coxeter systems of ranks \(0,1,2,3\), for which a complete effective classification and a solution of the isomorphism problem are obtained [2211.17038]. In the arithmetic framework based on the Tits representation, a small Coxeter group is one that admits a Coxeter system whose exponents all lie in \(\{1,2,3,\infty\}\); equivalently, it is a Coxeter group for which the Tits representation is integral [2212.09084]. These notions are distinct but complementary: the first is organized by rank and abstract isomorphism, while the second is organized by integrality, congruence subgroups, and arithmetic structure.

## 1. Terminology, presentations, and the two notions of smallness

A Coxeter system consists of a group \(W\) with a finite set \(S\) of generators and a symmetric Coxeter matrix \(M=(m_{ij})\) with \(m_{ii}=1\) and \(m_{ij}\in\{2,3,\dots,\infty\}\) for \(i\neq j\). Its standard presentation is
\[
W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,
\]
with the convention that \(m_{ij}=\infty\) means no relation. The complete Coxeter graph \(C(W,S)\) is the complete labeled graph on vertex set \(S\), and the rank is \(|S|\) [2211.17038].

Two current usages of “small Coxeter group” are standard in the cited literature.

| Sense | Criterion | Main setting |
|---|---|---|
| Rank-small | \(C(W,S)\) lies in \(G_0,G_1,G_2,G_3\) | Coxeter galaxy and isomorphism problem |
| Integral-small | some Coxeter system has all exponents in \(\{1,2,3,\infty\}\) | Tits representation and congruence theory |

In the first sense, “small” means ranks \(0\) through \(3\), and the focus is the geometry of the galaxy of Coxeter groups [2211.17038]. In the second sense, “small” means Tits-integral, and the focus is congruence subgroups, CSP, and crystallographic quotients [2510.23002].

A common source of confusion is that low rank and Tits-integrality are not the same condition. For example, \(H_3=\Delta(5,3,2)\) is a rank-\(3\) triangle Coxeter group in the galaxy sense, but \(H_3\) is listed among the finite irreducible types that are not small in the integral sense [2510.23002].

## 2. Classification in ranks \(0\) to \(3\)

The rank-small theory is completely explicit. Rank \(0\) consists of the trivial group \(1\), and rank \(1\) consists of \(C_2\) with presentation \(\langle s\mid s^2=1\rangle\) [2211.17038].

Every rank-\(2\) Coxeter group is dihedral:
\[
W=\langle s,t\mid s^2=t^2=1,\ (st)^m=1\rangle,
\qquad m\in\{2,3,\dots,\infty\}.
\]
For \(m<\infty\), this is the finite dihedral group \(D_{2m}\), also denoted \(I_2(m)\); for \(m=\infty\), it is the infinite dihedral group \(D_\infty\cong \mathbb Z\rtimes C_2\). Two rank-\(2\) Coxeter groups are isomorphic if and only if their parameters \(m\) agree [2211.17038]. In the finite reflection interpretation, \(I_2(m)\) is generated by reflections across two lines at angle \(\pi/m\), and \(|I_2(m)|=2m\); the special cases \(A_2=I_2(3)\), \(B_2=C_2=I_2(4)\), and \(G_2=I_2(6)\) are the standard crystallographic representatives [2601.02290].

Every rank-\(3\) Coxeter group is a triangle Coxeter group
\[
\Delta(p,q,r)=\langle a,b,c\mid a^2=b^2=c^2=1,\ (ab)^p=(bc)^q=(ca)^r=1\rangle,
\]
where \(p,q,r\in\{2,3,\dots,\infty\}\), again with the convention that exponent \(\infty\) means no relation. The isomorphism criterion is exact: \(\Delta(p,q,r)\cong \Delta(p',q',r')\) if and only if the multisets of edge labels agree,
\[
\{p,q,r\}=\{p',q',r'\}.
\]
The same theorem states that triangle Coxeter groups are profinitely rigid: isomorphism of profinite completions is equivalent to equality of the multisets \(\{p,q,r\}\) [2211.17038].

The geometric type of a triangle Coxeter group is controlled by the classical inequality
\[
\frac1p+\frac1q+\frac1r \gtrless 1.
\]
It is spherical if the sum is \(>1\), Euclidean if it is \(=1\), and hyperbolic if it is \(<1\). Canonical spherical examples are \(A_3=\Delta(3,3,2)\), \(B_3=\Delta(4,3,2)\), \(H_3=\Delta(5,3,2)\), and \(\Delta(2,2,n)\) for \(n<\infty\); Euclidean examples are \(\Delta(3,3,3)\), \(\Delta(4,4,2)\), \(\Delta(6,3,2)\), and \(\Delta(2,2,\infty)\); hyperbolic examples include \(\Delta(2,3,7)\), \(\Delta(2,5,5)\), \(\Delta(3,4,4)\), and the cusp cases \(\Delta(p,q,\infty)\) with \(1/p+1/q<1\) [2211.17038].

## 3. The galaxy of Coxeter groups in small ranks

The Coxeter galaxy \(G\) is a flag simplicial complex whose vertices are isomorphism classes of complete Coxeter graphs \(C(W,S)\). Two vertices are joined by an edge if the corresponding groups \(W\) are abstractly isomorphic, and layers \(G_k\) record rank \(k\) [2211.17038].

In small ranks the structure is especially rigid. The subcomplex \(G_{\le 3}\) is \(1\)-dimensional, has no \(2\)-simplices, and equals its vertical core. Rank \(0\) and rank \(1\) each contribute a single isolated vertex. In rank \(2\), all dihedral vertices are isolated except \(I_2(4k+2)\), which has a single vertical neighbor in rank \(3\). In rank \(3\), all triangle vertices are isolated; there are no horizontal edges inside \(G_3\) [2211.17038].

The only nontrivial adjacency in \(G_{\le 3}\) is
\[
I_2(4k+2)\;-\;\Delta(2k+1,2,2),
\qquad k\ge 1.
\]
This edge comes from a blow-up along a pseudo-transposition. In the formulation recalled by the paper, if a generator \(t\) lies in a maximal spherical parabolic of type \(I_2(4k+2)\) satisfying the pseudo-transposition conditions, one may blow up \((W,S)\) to a system \((W,S')\) with \(|S'|=|S|+1\) by adjoining the longest element of that parabolic and replacing \(t\) by a conjugate. In small ranks these blow-ups account for all vertical edges between \(G_2\) and \(G_3\) [2211.17038].

This yields a complete algorithmic picture. To decide isomorphism in \(G_{\le 3}\), one compares the dihedral parameter \(m\) in rank \(2\) and the unordered triple \(\{p,q,r\}\) in rank \(3\). A stronger statement is also proved: if \(C(W_1,S_1)\) and \(C(W_2,S_2)\) lie in \(G_{\le 3}\), then \(\widehat W_1\cong \widehat W_2\) implies \(W_1\cong W_2\). In small ranks, profinite completion is therefore a complete invariant [2211.17038].

## 4. Tits-integral small Coxeter groups

The arithmetic notion of smallness is defined through the Tits representation. Let \(V\) be the real vector space with basis \(\{e_i\}\), and define a symmetric bilinear form \(B\) by
\[
B(e_i,e_j)=
\begin{cases}
-\cos(\pi/m_{ij}), & m_{ij}\neq \infty,\\
-1, & m_{ij}=\infty.
\end{cases}
\]
For each \(i\), the Tits reflection is
\[
s_i(v)=v-2B(e_i,v)e_i.
\]
The resulting homomorphism \(\rho_{\mathrm{Tits}}:W\to GL(V)\) is faithful [2212.09084].

A Coxeter group is small in this sense if and only if the Tits representation is integral, and this happens if and only if all exponents satisfy \(m_{ij}\in\{1,2,3,\infty\}\) [2510.23002]. The point is that the only off-diagonal coefficients \(-2\cos(\pi/m_{ij})\) that are integral are precisely those arising from \(m_{ij}\in\{2,3,\infty\}\), with \(m_{ii}=1\) on the diagonal [2212.09084].

The classification statements quoted in the congruence literature are sharp. Among finite irreducible types, the small ones are \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\); in addition, \(I_2(6)\) and \(B_{2k+1}\) admit small Coxeter systems via direct product decompositions. The finite irreducible types that are not small are \(B_{2n}=C_{2n}\), \(I_2(5)\), \(I_2(n)\) with \(n=5\) or \(n\ge 7\), and the exceptional types \(F_4\), \(H_3\), \(H_4\) [2510.23002].

Among affine irreducibles, the small ones are \(\widetilde A_n\) for \(n\ge 2\), \(\widetilde D_n\) for \(n\ge 4\), \(\widetilde E_6\), \(\widetilde E_7\), \(\widetilde E_8\), and \(\widetilde A_1\). The remaining affine irreducible types \(\widetilde B_n\), \(\widetilde C_n\), \(\widetilde F_4\), and \(\widetilde G_2\) are not small [2510.23002].

## 5. Congruence subgroups, CSP, and affine structure

Once the Tits representation is integral, principal congruence subgroups are defined by
\[
W[m]=\ker\bigl(W\to GL_N(\mathbb Z)\to GL_N(\mathbb Z/m\mathbb Z)\bigr),
\qquad m\ge 2.
\]
The congruence subgroup property means that every finite-index subgroup contains \(W[m]\) for some \(m\) [2212.09084].

The negative direction was established first: an infinite small Coxeter group which is not virtually abelian does not have the congruence subgroup property [2212.09084]. The proof uses the existence of finite-index subgroups surjecting onto \(F_2\), together with the obstruction that a group with such a finite-index free image cannot have CSP for any integral representation [2510.23002].

The later completion of the picture is a positive theorem: if \(W\) is a small Coxeter group that is virtually abelian, equivalently each connected component of its Coxeter graph is spherical or affine, then \(W\) has CSP with respect to \(\rho_{\mathrm{Tits}}\) [2510.23002]. The two results together imply that, for small Coxeter groups, CSP holds if and only if the group is virtually abelian.

In irreducible affine small type, the structure of principal congruence subgroups is explicit. If \(W[\widetilde\Gamma]\) is irreducible affine and small, then \(W[\widetilde\Gamma][m]\le Q\) for \(m\ge 3\), where \(Q\) is the abelian translation subgroup, and \(W[\widetilde\Gamma][m]\) is a \(W[\Gamma]\)-invariant sublattice. For \(m=2\), \(W[\widetilde\Gamma][2]\) is an extension of such a \(W[\Gamma]\)-invariant sublattice by \(Z(W[\Gamma])\) [2510.23002].

The rank-\(1\) affine example \(\widetilde A_1\) is fully explicit:
\[
W[\widetilde A_1][m]=
\begin{cases}
\langle (s_0s_1)^m\rangle, & m\ \text{odd},\\
\langle (s_0s_1)^{m/2}\rangle, & m\ \text{even},
\end{cases}
\qquad m\ge 3.
\]
This coincides with the explicit description of the principal congruence subgroups of \(T_3\cong D_\infty\) in the earlier paper [2510.23002].

## 6. Examples, related families, and structural significance

The twin groups \(T_n\) and triplet groups \(L_n\) supply canonical infinite families of small Coxeter groups. They are defined by
\[
T_n=\langle s_1,\dots,s_{n-1}\mid s_k^2=1,\ s_is_j=s_js_i \text{ if } |i-j|\ge 2\rangle
\]
and
\[
L_n=\langle y_1,\dots,y_{n-1}\mid y_j^2=1,\ y_iy_{i+1}y_i=y_{i+1}y_iy_{i+1}\rangle.
\]
Both are Coxeter groups with all \(m_{ij}\in\{2,3,\infty\}\), hence small in the integral sense [2212.09084].

Their congruence theory is unusually concrete. The pure twin group satisfies
\[
PT_n=T_n[3]=T_n[6],
\]
and the pure triplet group satisfies
\[
PL_n=L_n[2].
\]
For \(n\ge 4\), both \(T_n\) and \(L_n\) fail CSP. By contrast, \(T_3\cong D_\infty\) has CSP, and its principal congruence subgroups admit the same odd/even formula displayed above [2212.09084].

These families also produce crystallographic quotients. The groups \(T_n/PT_n'\), \(T_n/T_n''\), and \(L_n/PL_n'\) are crystallographic groups, with dimensions
\[
\dim(T_n/PT_n')=2^{n-3}(n^2-5n+8)-1,
\]
\[
\dim(T_n/T_n'')=2n-5,
\]
and
\[
\dim(L_n/PL_n')=1+n!(2^{n-1}-1).
\]
The quotient \(T_n/T_n''\) moreover comes with an explicitly determined holonomy representation \(\vartheta:\mathbb Z_2^{n-1}\to GL_{2n-5}(\mathbb Z)\) whose matrices have entries in \(\{-1,0,1\}\) [2212.09084].

The broader significance of small Coxeter groups is therefore twofold. In low rank, they provide a setting where the galaxy completely resolves abstract isomorphism and even profinite rigidity [2211.17038]. In the Tits-integral setting, they define the natural domain for congruence subgroup theory, affine lattice structure, and crystallographic quotients [2510.23002]. A plausible implication is that the adjective “small” now names two complementary regimes: one where Coxeter groups are fully navigable by rank, and one where they are arithmetically tractable because their canonical linear representation is integral.

Source: https://www.emergentmind.com/topics/small-coxeter-groups