Small Coxeter Groups

• Small Coxeter groups are defined in two ways: as groups with low rank (0 to 3) and as groups whose Tits representations are integral with exponents in {1,2,3,∞}.
• The rank-small category offers an explicit classification of rank 0–3 groups using dihedral and triangle presentations to establish clear isomorphism criteria.
• Tits-integral small Coxeter groups facilitate detailed studies of congruence subgroups and crystallographic quotients, highlighting their arithmetic structure.

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 (Rego et al., 2022). 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 (Kumar et al., 2022). 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 $0,1,2,3$0 means no relation. The complete Coxeter graph $0,1,2,3$1 is the complete labeled graph on vertex set $0,1,2,3$2, and the rank is $0,1,2,3$3 (Rego et al., 2022).

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

Sense	Criterion	Main setting
Rank-small	$0,1,2,3$4 lies in $0,1,2,3$5	Coxeter galaxy and isomorphism problem
Integral-small	some Coxeter system has all exponents in $0,1,2,3$6	Tits representation and congruence theory

In the first sense, “small” means ranks $0,1,2,3$7 through $0,1,2,3$8, and the focus is the geometry of the galaxy of Coxeter groups (Rego et al., 2022). In the second sense, “small” means Tits-integral, and the focus is congruence subgroups, CSP, and crystallographic quotients (Kumar, 27 Oct 2025).

A common source of confusion is that low rank and Tits-integrality are not the same condition. For example, $0,1,2,3$9 is a rank-$\{1,2,3,\infty\}$0 triangle Coxeter group in the galaxy sense, but $\{1,2,3,\infty\}$1 is listed among the finite irreducible types that are not small in the integral sense (Kumar, 27 Oct 2025).

2. Classification in ranks $\{1,2,3,\infty\}$2 to $\{1,2,3,\infty\}$3

The rank-small theory is completely explicit. Rank $\{1,2,3,\infty\}$4 consists of the trivial group $\{1,2,3,\infty\}$5, and rank $\{1,2,3,\infty\}$6 consists of $\{1,2,3,\infty\}$7 with presentation $\{1,2,3,\infty\}$8 (Rego et al., 2022).

Every rank-$\{1,2,3,\infty\}$9 Coxeter group is dihedral: $W$0 For $W$1, this is the finite dihedral group $W$2, also denoted $W$3; for $W$4, it is the infinite dihedral group $W$5. Two rank-$W$6 Coxeter groups are isomorphic if and only if their parameters $W$7 agree (Rego et al., 2022). In the finite reflection interpretation, $W$8 is generated by reflections across two lines at angle $W$9, and $S$0; the special cases $S$1, $S$2, and $S$3 are the standard crystallographic representatives (Baez, 5 Jan 2026).

Every rank-$S$4 Coxeter group is a triangle Coxeter group

$S$5

where $S$6, again with the convention that exponent $S$7 means no relation. The isomorphism criterion is exact: $S$8 if and only if the multisets of edge labels agree,

$S$9

The same theorem states that triangle Coxeter groups are profinitely rigid: isomorphism of profinite completions is equivalent to equality of the multisets $M=(m_{ij})$0 (Rego et al., 2022).

The geometric type of a triangle Coxeter group is controlled by the classical inequality

$M=(m_{ij})$1

It is spherical if the sum is $M=(m_{ij})$2, Euclidean if it is $M=(m_{ij})$3, and hyperbolic if it is $M=(m_{ij})$4. Canonical spherical examples are $M=(m_{ij})$5, $M=(m_{ij})$6, $M=(m_{ij})$7, and $M=(m_{ij})$8 for $M=(m_{ij})$9; Euclidean examples are $m_{ii}=1$0, $m_{ii}=1$1, $m_{ii}=1$2, and $m_{ii}=1$3; hyperbolic examples include $m_{ii}=1$4, $m_{ii}=1$5, $m_{ii}=1$6, and the cusp cases $m_{ii}=1$7 with $m_{ii}=1$8 (Rego et al., 2022).

3. The galaxy of Coxeter groups in small ranks

The Coxeter galaxy $m_{ii}=1$9 is a flag simplicial complex whose vertices are isomorphism classes of complete Coxeter graphs $m_{ij}\in\{2,3,\dots,\infty\}$0. Two vertices are joined by an edge if the corresponding groups $m_{ij}\in\{2,3,\dots,\infty\}$1 are abstractly isomorphic, and layers $m_{ij}\in\{2,3,\dots,\infty\}$2 record rank $m_{ij}\in\{2,3,\dots,\infty\}$3 (Rego et al., 2022).

In small ranks the structure is especially rigid. The subcomplex $m_{ij}\in\{2,3,\dots,\infty\}$4 is $m_{ij}\in\{2,3,\dots,\infty\}$5-dimensional, has no $m_{ij}\in\{2,3,\dots,\infty\}$6-simplices, and equals its vertical core. Rank $m_{ij}\in\{2,3,\dots,\infty\}$7 and rank $m_{ij}\in\{2,3,\dots,\infty\}$8 each contribute a single isolated vertex. In rank $m_{ij}\in\{2,3,\dots,\infty\}$9, all dihedral vertices are isolated except $i\neq j$0, which has a single vertical neighbor in rank $i\neq j$1. In rank $i\neq j$2, all triangle vertices are isolated; there are no horizontal edges inside $i\neq j$3 (Rego et al., 2022).

The only nontrivial adjacency in $i\neq j$4 is

$i\neq j$5

This edge comes from a blow-up along a pseudo-transposition. In the formulation recalled by the paper, if a generator $i\neq j$6 lies in a maximal spherical parabolic of type $i\neq j$7 satisfying the pseudo-transposition conditions, one may blow up $i\neq j$8 to a system $i\neq j$9 with $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$0 by adjoining the longest element of that parabolic and replacing $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$1 by a conjugate. In small ranks these blow-ups account for all vertical edges between $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$2 and $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$3 (Rego et al., 2022).

This yields a complete algorithmic picture. To decide isomorphism in $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$4, one compares the dihedral parameter $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$5 in rank $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$6 and the unordered triple $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$7 in rank $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$8. A stronger statement is also proved: if $$W=\langle s_1,\dots,s_n \mid s_i^2=1,\ (s_is_j)^{m_{ij}}=1 \text{ for } i\neq j \text{ with } m_{ij}<\infty\rangle,$$9 and $0,1,2,3$00 lie in $0,1,2,3$01, then $0,1,2,3$02 implies $0,1,2,3$03. In small ranks, profinite completion is therefore a complete invariant (Rego et al., 2022).

4. Tits-integral small Coxeter groups

The arithmetic notion of smallness is defined through the Tits representation. Let $0,1,2,3$04 be the real vector space with basis $0,1,2,3$05, and define a symmetric bilinear form $0,1,2,3$06 by

$0,1,2,3$07

For each $0,1,2,3$08, the Tits reflection is

$0,1,2,3$09

The resulting homomorphism $0,1,2,3$10 is faithful (Kumar et al., 2022).

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 $0,1,2,3$11 (Kumar, 27 Oct 2025). The point is that the only off-diagonal coefficients $0,1,2,3$12 that are integral are precisely those arising from $0,1,2,3$13, with $0,1,2,3$14 on the diagonal (Kumar et al., 2022).

The classification statements quoted in the congruence literature are sharp. Among finite irreducible types, the small ones are $0,1,2,3$15, $0,1,2,3$16, $0,1,2,3$17, $0,1,2,3$18, $0,1,2,3$19; in addition, $0,1,2,3$20 and $0,1,2,3$21 admit small Coxeter systems via direct product decompositions. The finite irreducible types that are not small are $0,1,2,3$22, $0,1,2,3$23, $0,1,2,3$24 with $0,1,2,3$25 or $0,1,2,3$26, and the exceptional types $0,1,2,3$27, $0,1,2,3$28, $0,1,2,3$29 (Kumar, 27 Oct 2025).

Among affine irreducibles, the small ones are $0,1,2,3$30 for $0,1,2,3$31, $0,1,2,3$32 for $0,1,2,3$33, $0,1,2,3$34, $0,1,2,3$35, $0,1,2,3$36, and $0,1,2,3$37. The remaining affine irreducible types $0,1,2,3$38, $0,1,2,3$39, $0,1,2,3$40, and $0,1,2,3$41 are not small (Kumar, 27 Oct 2025).

5. Congruence subgroups, CSP, and affine structure

Once the Tits representation is integral, principal congruence subgroups are defined by

$0,1,2,3$42

The congruence subgroup property means that every finite-index subgroup contains $0,1,2,3$43 for some $0,1,2,3$44 (Kumar et al., 2022).

The negative direction was established first: an infinite small Coxeter group which is not virtually abelian does not have the congruence subgroup property (Kumar et al., 2022). The proof uses the existence of finite-index subgroups surjecting onto $0,1,2,3$45, together with the obstruction that a group with such a finite-index free image cannot have CSP for any integral representation (Kumar, 27 Oct 2025).

The later completion of the picture is a positive theorem: if $0,1,2,3$46 is a small Coxeter group that is virtually abelian, equivalently each connected component of its Coxeter graph is spherical or affine, then $0,1,2,3$47 has CSP with respect to $0,1,2,3$48 (Kumar, 27 Oct 2025). 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 $0,1,2,3$49 is irreducible affine and small, then $0,1,2,3$50 for $0,1,2,3$51, where $0,1,2,3$52 is the abelian translation subgroup, and $0,1,2,3$53 is a $0,1,2,3$54-invariant sublattice. For $0,1,2,3$55, $0,1,2,3$56 is an extension of such a $0,1,2,3$57-invariant sublattice by $0,1,2,3$58 (Kumar, 27 Oct 2025).

The rank-$0,1,2,3$59 affine example $0,1,2,3$60 is fully explicit: $0,1,2,3$61 This coincides with the explicit description of the principal congruence subgroups of $0,1,2,3$62 in the earlier paper (Kumar, 27 Oct 2025).

6. Examples, related families, and structural significance

The twin groups $0,1,2,3$63 and triplet groups $0,1,2,3$64 supply canonical infinite families of small Coxeter groups. They are defined by

$0,1,2,3$65

and

$0,1,2,3$66

Both are Coxeter groups with all $0,1,2,3$67, hence small in the integral sense (Kumar et al., 2022).

Their congruence theory is unusually concrete. The pure twin group satisfies

$0,1,2,3$68

and the pure triplet group satisfies

$0,1,2,3$69

For $0,1,2,3$70, both $0,1,2,3$71 and $0,1,2,3$72 fail CSP. By contrast, $0,1,2,3$73 has CSP, and its principal congruence subgroups admit the same odd/even formula displayed above (Kumar et al., 2022).

These families also produce crystallographic quotients. The groups $0,1,2,3$74, $0,1,2,3$75, and $0,1,2,3$76 are crystallographic groups, with dimensions

$0,1,2,3$77

$0,1,2,3$78

and

$0,1,2,3$79

The quotient $0,1,2,3$80 moreover comes with an explicitly determined holonomy representation $0,1,2,3$81 whose matrices have entries in $0,1,2,3$82 (Kumar et al., 2022).

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 (Rego et al., 2022). In the Tits-integral setting, they define the natural domain for congruence subgroup theory, affine lattice structure, and crystallographic quotients (Kumar, 27 Oct 2025). 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.