Even Coxeter Groups: Structure & Rigidity

• Even Coxeter groups are defined by having all finite off-diagonal entries in the Coxeter matrix as even, which establishes a clear structural subclass including right-angled groups.
• Their evenness induces strong rigidity in diagram structure and self-similarity, underpinning results in first-order model theory and automorphism invariance.
• The parity condition facilitates computational efficiency, enabling logspace computation of geodesic normal forms and invariant Parikh images for geodesic words.

Searching arXiv for recent and foundational papers on even Coxeter groups. An even Coxeter group is a Coxeter group admitting a finite Coxeter system $(W,S)$ in which every finite off-diagonal entry of the Coxeter matrix is even. In standard form,

$$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$

with $m_S(s,s)=1$, $m_S(s,t)=m_S(t,s)$, and $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$ for $s\neq t$ in the even case (Muhlherr et al., 2020, André et al., 2024). Even Coxeter groups include right-angled Coxeter groups, where the only finite labels are $2$, and they occupy a particularly rigid position in several parts of contemporary Coxeter theory: first-order model theory, algorithmic complexity, geodesic growth, separability, automorphism theory, and exponential growth (André et al., 2024, Diekert et al., 2012, Caprace et al., 2012, Bossart et al., 2 Jul 2026).

1. Definition, subclasses, and diagrammatic structure

A Coxeter system $(W,S)$ is even when all finite entries $m_{st}$ with $s\neq t$ are even; equivalently, every defining relation $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$0 has even length $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$1 (Muhlherr et al., 2020, Diekert et al., 2012). In the associated Coxeter diagram, every finite edge label is therefore an even integer, while $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$2 indicates the absence of a finite-order relation. Right-angled Coxeter groups form a distinguished subclass with $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$3 for $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$4 (André et al., 2024, Diekert et al., 2012).

Several structural adjectives recur in the even setting. A system is 2-spherical if every rank-2 parabolic $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$5 is finite, equivalently $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$6 for all distinct $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$7; it is irreducible if the Coxeter diagram is connected; it is spherical if $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$8 is finite; and it is affine if $$W=\left\langle S \mid (st)^{m_S(s,t)}=1 \text{ whenever } m_S(s,t)<\infty \right\rangle,$$9 is infinite and admits an affine reflection representation (Muhlherr et al., 2020). In the hyperbolic direction, the 2024 rigidity work uses Gromov-hyperbolic Coxeter groups and recalls Moussong’s criterion: a Coxeter group is hyperbolic iff it does not contain $m_S(s,s)=1$0 and there is no affine parabolic subgroup of rank $m_S(s,s)=1$1 (André et al., 2024).

The even hypothesis has diagrammatic consequences beyond parity. For even Coxeter groups, there is a distinguished even Coxeter system $m_S(s,s)=1$2, and recent work proves that among all Coxeter systems generating $m_S(s,s)=1$3, this unique even Coxeter system realizes the minimal exponential growth (Bossart et al., 2 Jul 2026). In a different direction, for finite-rank even Coxeter groups the absence of a $m_S(s,s)=1$4-triangle—equivalently, in this setting, the absence of affine subdiagrams of rank $m_S(s,s)=1$5—is the geometric condition that controls cocompactness of the Niblo–Reeves cubulation and underlies conjugacy separability (Caprace et al., 2012).

2. Reflections, parabolics, and even-specific rigidity mechanisms

For a Coxeter system $m_S(s,s)=1$6, the reflections are the conjugates of the simple generators,

$m_S(s,s)=1$7

and an $m_S(s,s)=1$8-reflection subgroup is a subgroup generated by its intersection with $m_S(s,s)=1$9 (Muhlherr et al., 2020). In the even case, reflections interact especially well with endomorphisms preserving the Coxeter combinatorics. An $m_S(s,t)=m_S(t,s)$0-self-similarity is an endomorphism $m_S(s,t)=m_S(t,s)$1 such that $m_S(s,t)=m_S(t,s)$2 for every $m_S(s,t)=m_S(t,s)$3, and

$m_S(s,t)=m_S(t,s)$4

for all $m_S(s,t)=m_S(t,s)$5 (Muhlherr et al., 2020, André et al., 2024). For finite-rank even Coxeter systems, such self-similarities are rigid: if $m_S(s,t)=m_S(t,s)$6, then $m_S(s,t)=m_S(t,s)$7 is a Coxeter system and $m_S(s,t)=m_S(t,s)$8 is an isomorphism (Muhlherr et al., 2020). The 2024 rigidity paper records the same phenomenon as an even-case fact originating in work of Mühlherr and collaborators (André et al., 2024).

Standard parabolic subgroups also behave particularly cleanly in even Coxeter groups. For $m_S(s,t)=m_S(t,s)$9, the standard parabolic subgroup is $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$0. In the even case there is a canonical retraction

$m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$1

and these retractions commute for different $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$2 (André et al., 2024). This commuting-retraction formalism is specific to the even setting in the cited account, and it is used to control finite special subgroups, spherical substructures, and first-order reconstruction arguments (André et al., 2024).

These structural devices connect local diagram data with global rigidity. In particular, they allow tuples of reflections with the same rank-2 orders as a Coxeter basis to be recognized as images of the standard basis under self-similarities, and in the even case those self-similarities are automatically isomorphisms (Muhlherr et al., 2020).

3. First-order model theory and logical rigidity

Even Coxeter groups are one of the principal rigid classes in the first-order model theory of Coxeter groups. A central result is that if $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$3 and $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$4 are even Coxeter groups and $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$5, then $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$6; moreover, $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$7 is an algebraically prime model of its first-order theory, meaning that it embeds in every model of its theory (André et al., 2024). In the terminology introduced there, this places even Coxeter groups among the strongest positive instances of first-order rigidity available for Coxeter groups.

The hyperbolic even case is stronger still. Hyperbolic even Coxeter groups are AE-torsion-rigid: if $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$8 is a hyperbolic even Coxeter group and $m_S(s,t)\in\{2,4,6,\dots\}\cup\{\infty\}$9 is a finitely torsion-generated group that is AE-equivalent to $s\neq t$0, then $s\neq t$1 (André et al., 2024). This is significant because general Coxeter groups do not enjoy such rigidity among finitely generated groups: the same paper recalls Sela’s phenomenon that $s\neq t$2 is elementarily equivalent to $s\neq t$3 for any nonabelian free group $s\neq t$4 (André et al., 2024).

A more specialized but sharper theorem holds for irreducible, finite-rank, 2-spherical even non-affine Coxeter groups. In that class, the set of reflections is definable without parameters in the pure group language, the group is a prime model of its theory, and elementary equivalence is equivalent to isomorphism of Coxeter diagrams: $s\neq t$5 for irreducible, 2-spherical, even, non-affine Coxeter groups $s\neq t$6 (Muhlherr et al., 2020). The proof combines parameter-free definability of reflections, strong rigidity of 2-spherical Coxeter groups, and the even-case theorem that self-similarities are isomorphisms onto reflection subgroups (Muhlherr et al., 2020).

The stability-theoretic picture is less complete. The 2020 model-theoretic study gives a near-classification of superstable finite-rank Coxeter groups, essentially the affine ones, but explicitly leaves open the general 2-spherical non-affine case; this unresolved block includes 2-spherical even non-affine groups (Muhlherr et al., 2020). Thus even Coxeter groups exhibit strong primeness and diagram rigidity without a corresponding complete superstability classification.

4. Automaticity, normal forms, and low-space computation

All Coxeter groups are biautomatic, and therefore every even Coxeter group is biautomatic (Osajda et al., 2022). The biautomatic structure is built from the voracious language $s\neq t$7, a regular geodesic language defined using walls in the Cayley graph and the Davis complex, a partial order $s\neq t$8 on elements lying on geodesics, and the voracious projection $s\neq t$9, the maximal element of a certain bounded predecessor set $2$0 (Osajda et al., 2022). The resulting language is regular and satisfies both left and right fellow-traveler properties, yielding biautomaticity uniformly across the class of Coxeter groups (Osajda et al., 2022).

Evenness becomes decisive in complexity questions about geodesics. For arbitrary Coxeter groups, the geodesic length of an input word and the set of letters occurring in any geodesic representative can be computed in deterministic logspace (Diekert et al., 2012). In even Coxeter groups, one obtains a stronger invariant: if $2$1 and $2$2 are geodesics representing the same group element, then

$2$3

so all geodesic representatives have the same Parikh-image (Diekert et al., 2012). Consequently, in an even Coxeter group the Parikh-image of the shortlex normal form can be computed in logspace (Diekert et al., 2012).

The mechanism is specific to parity. In the even case, the relevant braid moves between geodesics preserve the multiplicity of each generator, so the Parikh-image is a well-defined invariant of the represented element (Diekert et al., 2012). This does not extend in the same form to arbitrary Coxeter groups with odd labels. For right-angled Coxeter groups, which are a subclass of even Coxeter groups, the shortlex normal form itself is computable in logspace (Diekert et al., 2012).

5. Geodesic growth and local combinatorics

Geodesic growth for even Coxeter groups can be governed by coarse local data rather than global isomorphism type. For a finitely generated group $2$4, the geodesic growth function counts geodesic words of length $2$5, and the geodesic growth series is

$2$6

(Antolín et al., 2012). In triangle-free, star-regular even Coxeter systems, geodesic growth depends only on the number of generators $2$7 and the isomorphism class of the labeled star $2$8 of a vertex (Antolín et al., 2012).

The proof uses a forbidden-word description of the geodesic language. For an even Coxeter group and $2$9, the centralizer $(W,S)$0 is generated by $(W,S)$1 together with explicitly described words $(W,S)$2 indexed by neighbors $(W,S)$3 of $(W,S)$4, and the geodesic language is exactly the language avoiding forbidden subwords of the form

$(W,S)$5

where $(W,S)$6 is a geodesic word on the alphabet $(W,S)$7 (Antolín et al., 2012). Grigorchuk–Nagnibeda rigid-chain technology then reduces geodesic growth to counts of locally determined overlaps of these forbidden words (Antolín et al., 2012).

This local determination has explicit consequences. The paper exhibits non-isomorphic triangle-free, star-regular even Coxeter groups with identical geodesic growth: one example compares the union of two squares with alternating labels $(W,S)$8 to a single octagon with the same alternating labels. By Bahls’ isomorphism results for even Coxeter groups they are non-isomorphic, but their geodesic growth series coincide (Antolín et al., 2012). In this class, geodesic growth is therefore a comparatively weak invariant.

6. Conjugacy separability, automorphisms, and exponential growth

Even Coxeter groups also support strong separability and automorphism-rigidity theorems. If $(W,S)$9 is a finite-rank even Coxeter group whose Coxeter diagram has no $m_{st}$0-triangles, then $m_{st}$1 is conjugacy separable (Caprace et al., 2012). This applies in particular to all right-angled Coxeter groups and all word-hyperbolic even Coxeter groups of finite rank (Caprace et al., 2012). The proof combines cocompact cubulation, heredity of conjugacy separability in a finite-index subgroup, and the even-case system of commuting retractions onto standard parabolic subgroups (Caprace et al., 2012).

Automorphism theory is correspondingly rigid. For finitely generated Coxeter groups, an automorphism preserving all conjugacy classes of elements that can be written as products of pairwise distinct generators is inner; equivalently stated in a special case, every pointwise inner automorphism of a finitely generated Coxeter group is inner (Caprace et al., 2012). Under the conjugacy-separable even hypotheses above, this yields residual finiteness of $m_{st}$2 (Caprace et al., 2012).

Recent growth theory adds another canonical feature of the even structure. For a finitely generated group with generating set $m_{st}$3, the exponential growth rate is

$m_{st}$4

and for a Coxeter group $m_{st}$5 one may take the infimum over Coxeter generating sets, denoted $m_{st}$6 (Bossart et al., 2 Jul 2026). If $m_{st}$7 is an even Coxeter group, then

$m_{st}$8

where $m_{st}$9 is the unique even Coxeter system (Bossart et al., 2 Jul 2026). The proof runs Mihalik’s algorithm from an arbitrary Coxeter system to the even one and shows that diagram twisting preserves growth while blow-downs along pseudo-transpositions do not decrease it (Bossart et al., 2 Jul 2026). Thus the even Coxeter system is not only algebraically distinguished; among Coxeter presentations, it is also growth-minimizing.

Taken together, these results present even Coxeter groups as a class in which parity assumptions force unusually strong coherence between the Coxeter diagram, reflection structure, first-order theory, normal-form theory, and growth. Their defining evenness controls self-similarities, enables commuting parabolic retractions, stabilizes Parikh data of geodesics, and often allows the full Coxeter diagram to be recovered from logical or asymptotic invariants (Muhlherr et al., 2020, André et al., 2024, Diekert et al., 2012, Bossart et al., 2 Jul 2026).