Books

• Invitation au débat mathématique
  T. Gilbert et T. Marquis (dir.), et H. Ben Aïcha, I. Berlanger, C. Mousset, L. Ninove et D. Zimmer,
  Presses Universitaires de Louvain (2025), xvi+432 pp., ISBN 978-2-39061-590-0. PUL | page compagnon | résumé

  Résumé:

  Et si on incitait les élèves à débattre entre eux de situations mathématiques ? Dans un monde où l’esprit critique et l’autonomie de pensée sont essentiels, cet ouvrage invite à développer une culture du débat mathématique dans les classes. À partir d’une question mathématique, les élèves sont amenés à communiquer leurs idées, écouter celles des autres, argumenter et prendre du recul sur leurs raisonnements. Ce dispositif les place dans une posture active face aux apprentissages tout en promouvant le développement de compétences citoyennes au cours de mathématiques.
  
  Invitation au débat mathématique propose aux acteurs de l’enseignement des outils concrets pour intégrer cette pratique dans les classes, de la fin du primaire au supérieur. Il offre à la fois un guide synthétique pour la mise en oeuvre de débats et un recueil de 50 énoncés commentés destinés à provoquer le débat avec des élèves de tous âges. Les réflexions et analyses sont abondamment illustrées d’exemples, de témoignages d’enseignants et d’échos des classes.
  
  Que la pratique des débats mathématiques vous soit encore inconnue ou déjà familière, cet ouvrage nourrira votre réflexion au service d’un objectif fondamental : que les élèves pensent les mathématiques par eux-mêmes !
  
  Thérèse Gilbert (EPHEC-Éducation), Timothée Marquis (FNRS & UCLouvain), Habib Ben Aïcha (Athénée Royal de Jette), Isabelle Berlanger (EPHEC-Éducation), Céline Mousset (HELHa), Laure Ninove (UCLouvain & HE Vinci) et Daniel Zimmer sont tous les sept membres du Groupe d’enseignement mathématique (GEM). Ce groupe, créé en 1977 par Nicolas Rouche à l’UCLouvain, rassemble une trentaine d’acteurs de l’enseignement à tous niveaux autour d’une vision commune : mettre le sens et la réflexion au coeur de l’apprentissage des mathématiques.

  | extraits

  Extraits:

  Voici quelques extraits du livre: la table des matières, l'avant-propos, l'introduction, un extrait du chapitre 1, un exemple de fiche de débat, ainsi que des tables offrant une vue d’ensemble des 50 fiches et énoncés supplémentaires en fonction de l’âge du public d’élèves visé, ainsi que du domaine mathématique sur lequel porte le débat.



• An introduction to Kac-Moody groups over fields
  EMS Textbooks in Mathematics (2018), European Mathematical Society (EMS), Zürich, xi+331 p., ISBN 978-3-03719-187-3. doi | corrections | abstract

  Abstract:

  The interest for Kac-Moody algebras and groups has grown exponentially in the past decades, both in the mathematical and physics communities, and with it also the need of an introductory textbook on the topic.
  
  The aims of this book are twofold:

  • to offer an accessible, reader-friendly and self-contained introduction to Kac-Moody algebras and groups;
  • to clean the foundations and to provide a unified treatment of the theory.

  
  The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac-Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac-Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey.
  
  The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested in learning about Kac-Moody algebras and/or groups, from graduate (master) students to specialists.

  | minicourse

  Overview of the book:

  I gave a concise overview of the content of the book during a minicourse at the workshop ``Kac-Moody geometry'' in July 2023 in Kiel. Before the workshop, I prepared a short "Prerequisites" note, mainly for participants unfamiliar with Kac-Moody algebras. The minicourse itself consisted in three $1$-hour lectures, and is available here.

Publications and preprints

• Stable reflection length in Coxeter groups (with Francesco Fournier-Facio and Marco Lotz)
  Preprint (2025). pdf | doi | arXiv | abstract

  Abstract:

  We introduce stable reflection length in Coxeter groups, as a way to study the asymptotic behaviour of reflection length. This creates connections to other well-studied stable length functions in groups, namely stable commutator length and stable torsion length. As an application, we give a complete characterisation of elements whose reflection length is unbounded on powers.



• Amalgams of rational unipotent groups and residual nilpotence (with Pierre-Emmanuel Caprace)
  Preprint (2024). pdf | doi | arXiv | abstract

  Abstract:

  We provide sufficient conditions for a free amalgamated product of torsionfree nilpotent groups to be residually nilpotent. We also characterise the residual nilpotence of certain higher-dimensional amalgams of unipotent groups over the rationals (known as KMS groups) in terms of their defining Cartan matrix. As an application, we give a normal form for the elements of a minimal Kac-Moody group over the rationals.



• On the centre of Iwahori-Hecke algebras (with Sven Raum)
  Journal of Algebra 656 (2024), 276-293. pdf | doi | arXiv | abstract

  Abstract:

  We prove triviality of the centre of arbitrary Hecke algebras of irreducible non-finite non-affine type. This result is obtained as a consequence of the following structure result for conjugacy classes of the underlying Coxeter groups. If $W$ is any infinite irreducible Coxeter group and $w \in W$ is a nontrivial element that is assumed not be a translation in case $W$ is affine, then there is an infinite sequence of conjugates of $w$ by Coxeter generators whose length is non-decreasing and tends to infinity.



• Locally normal subgroups and ends of locally compact Kac-Moody groups (with Pierre-Emmanuel Caprace and Colin D. Reid)
  Münster J. Math. 15 (2022), no. 2, 473–498. pdf | doi | arXiv | abstract

  Abstract:

  A locally normal subgroup in a topological group is a subgroup whose normaliser is open. In this paper, we provide a detailed description of the large-scale structure of closed locally normal subgroups of complete Kac-Moody groups over finite fields. Combining that description with the main result from [CMR21], we show that under mild assumptions, if the Kac-Moody group is one-ended (a property that is easily determined from the generalised Cartan matrix), then it is locally indecomposable, which means that no open subgroup decomposes as a nontrivial direct product.



• Growing trees from compact subgroups (with Pierre-Emmanuel Caprace and Colin D. Reid)
  Groups Geom. Dyn. 18 (2024), no. 1, pp. 327–352. pdf | doi | arXiv | abstract

  Abstract:

  We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups $G$, finding a sufficient condition for $G$ to have more than one end in terms of its compact subgroups. The condition actually results in an action of $G/N$ on a tree with faithful micro-supported action on the boundary, where $N$ is compact, and is closely related to the Boolean algebra formed by the centralisers of locally normal subgroups of $G/N$. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group $G$, for all direct factors of open subgroups of $G$ to be trivial or open.



• Structure of conjugacy classes in Coxeter groups
  Astérisque No 457 (2025), viii+135 pp., ISBN 978-2-37905-201-9. pdf | doi | arXiv | GAP | abstract

  Abstract:

  This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts.
  
  Let $(W,S)$ be a Coxeter system. A cyclic shift of an element $w\in W$ is a conjugate of $w$ of the form $sws$ for some simple reflection $s\in S$ such that $\ell_S(sws)\leq\ell_S(w)$. The cyclic shift class of $w$ is then the set of elements of $W$ that can be obtained from $w$ by a sequence of cyclic shifts. Given a subset $K\subseteq S$ such that $W_K:=\langle K\rangle\subseteq W$ is finite, we also call two elements $w,w'\in W$ $K$-conjugate if $w,w'$ normalise $W_K$ and $w'=w_0(K)ww_0(K)$, where $w_0(K)$ is the longest element of $W_K$.
  
  Let $\mathcal O$ be a conjugacy class in $W$, and let $\mathcal O^{\min}$ be the set of elements of minimal length in $\mathcal O$. Then $\mathcal O^{\min}$ is the disjoint union of finitely many cyclic shift classes $C_1,\dots,C_k$. We define the structural conjugation graph associated to $\mathcal O$ to be the graph with vertices $C_1,\dots,C_k$, and with an edge between distinct vertices $C_i,C_j$ if they contain representatives $u\in C_i$ and $v\in C_j$ such that $u,v$ are $K$-conjugate for some $K\subseteq S$.
  
  In this paper, we compute explicitely the structural conjugation graph associated to any (possibly twisted) conjugacy class in $W$, and show in particular that it is connected (that is, any two conjugate elements of $W$ differ only by a sequence of cyclic shifts and $K$-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element $w\in W$, as well as the existence of natural decompositions of $w$ as a product of a ``torsion part'' and of a ``straight part'', with useful properties.



• Hyperfiniteness of boundary actions of hyperbolic groups (with Marcin Sabok)
  Mathematische Annalen 377 (2020), no. 3-4, 1129-1153. pdf | doi | arXiv | abstract

  Abstract:

  We prove that for every finitely generated hyperbolic group $G$, the action of $G$ on its Gromov boundary induces a hyperfinite equivalence relation.



• Cyclically reduced elements in Coxeter groups
  Annales Scientifiques de l'École normale supérieure 54 (2021), Fasc. 2, pp. 483-502. pdf | doi | arXiv | abstract

  Abstract:

  Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.



• On the structure of Kac-Moody algebras
  Canadian Journal of Mathematics Volume 73, Issue 4 (2021), pp. 1124-1152. pdf | doi | arXiv | abstract

  Abstract:

  Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y\in\mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$.



• Closed sets of real roots in Kac-Moody root systems (with Pierre-Emmanuel Caprace)
  preprint (2018). pdf | doi | arXiv | abstract

  Abstract:

  In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.



• On geodesic ray bundles in buildings
  Geom. Dedicata 202 Nr. 1 (2018), 27-43. pdf | doi | arXiv | abstract

  Abstract:

  Let $X$ be a building, identified with its Davis realisation. In this paper, we provide for each $x\in X$ and each $\eta$ in the visual boundary $\partial X$ of $X$ a description of the geodesic ray bundle $\mathrm{Geo}(x,\eta)$, namely, of the reunion of all combinatorial geodesic rays (corresponding to infinite minimal galleries in the chamber graph of $X$) starting from $x$ and pointing towards $\eta$. When $X$ is locally finite and hyperbolic, we show that the symmetric difference between $\mathrm{Geo}(x,\eta)$ and $\mathrm{Geo}(y,\eta)$ is always finite, for $x,y\in X$ and $\eta\in\partial X$. This gives a positive answer to a question of Huang, Sabok and Shinko in the setting of buildings. Combined with their results and a construction of Bourdon, our results then yield examples of hyperbolic groups $G$ with Kazhdan's property (T) such that the $G$-action on its Gromov boundary is hyperfinite.



• Half-Lie groups (with Karl-Hermann Neeb)
  Transform. Groups 23 (2018), no. 3, 801-840. pdf | doi | arXiv | abstract

  Abstract:

  In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If $G$ and $N$ are Banach-Lie groups and $\pi : G \to \mathrm{Aut}(N)$ is a homomorphism defining a continuous action of $G$ on $N$, then $H := N \rtimes_\pi G$ is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not to be. We show that these groups share surprisingly many properties with Banach-Lie groups: (a) for every regulated function $\xi : [0,1] \to T_1H$ the initial value problem $\dot\gamma(t) = \gamma(t)\xi(t)$, $\gamma(0)= 1_H$, has a solution and the corresponding evolution map from curves in $T_1H$ to curves in $H$ is continuous; (b) every $C^1$-curve $\gamma$ with $\gamma(0) = 1$ and $\gamma'(0) = x$ satisfies $\lim_{n \to \infty} \gamma(t/n)^n = \exp(tx)$; (c) the Trotter formula holds for $C^1$ one-parameter groups in $H$; (d) the subgroup $N^\infty$ of elements with smooth $G$-orbit maps in $N$ carries a natural Fréchet-Lie group structure for which the $G$-action is smooth; (e) the resulting Fréchet-Lie group $H^\infty := N^\infty \rtimes G$ is also regular in the sense of (a).



• Positive energy representations of double extensions of Hilbert loop algebras (with Karl-Hermann Neeb)
  J. Math. Soc. Japan 69 Nr. 4 (2017), 1485-1518. pdf | doi | arXiv | abstract

  Abstract:

  A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.



• Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity
  Annales de l'Institut Fourier, Volume 69 (2019) no. 6, p. 2519-2576. pdf | doi | arXiv | abstract

  Abstract:

  Let $k$ be a field and $A$ be a generalised Cartan matrix, and let ${\mathfrak G}_A(k)$ be the corresponding minimal Kac-Moody group of simply connected type over $k$. Consider the completion ${\mathfrak G}_A^{pma}(k)$ of ${\mathfrak G}_A(k)$ introduced by O. Mathieu and G. Rousseau, and let ${\mathfrak U}_A^{ma+}(k)$ denote the unipotent radical of the positive Borel subgroup of ${\mathfrak G}_A^{pma}(k)$. In this paper, we exhibit some functoriality dependence of the groups ${\mathfrak U}_A^{ma+}(k)$ and ${\mathfrak G}_A^{pma}(k)$ on their Lie algebra. We also produce a large class of examples of minimal Kac-Moody groups ${\mathfrak G}_A(k)$ that are not dense in their Mathieu-Rousseau completion ${\mathfrak G}_A^{pma}(k)$. Finally, we explain how the problematic of providing a unified theory of complete Kac-Moody groups is related to the conjecture of Gabber-Kac simplicity of ${\mathfrak G}_A^{pma}(k)$, stating that every normal subgroup of ${\mathfrak G}_A^{pma}(k)$ that is contained in ${\mathfrak U}_A^{ma+}(k)$ must be trivial. We present several motivations for the study of this conjecture, as well as several applications of our functoriality theorem, with contributions to the question of (non-)linearity of ${\mathfrak U}_A^{ma+}(k)$, and to the isomorphism problem for complete Kac-Moody groups over finite fields. For $k$ finite, we also make some observations on the structure of ${\mathfrak U}_A^{ma+}(k)$ in the light of some important concepts from the theory of pro-$p$ groups.



• Isomorphisms of twisted Hilbert loop algebras (with Karl-Hermann Neeb)
  Canad. J. Math. 69 (2017), 453-480. pdf | doi | arXiv | abstract

  Abstract:

  The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.



• Positive energy representations for locally finite split Lie algebras (with Karl-Hermann Neeb)
  Int. Math. Res. Notices (2016); 2016 (21): 6689-6712. pdf | doi | arXiv | abstract

  Abstract:

  Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with $\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary highest weight representation $(\rho_\lambda, V^\lambda)$ of $\mathfrak g$ extends to a representation $\tilde\rho_\lambda$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tilde\rho_\lambda$ is a positive energy representation if the spectrum of $-i\tilde\rho_\lambda(D)$ is bounded from below. In the present note we characterise all pairs $(\lambda,D)$ with $\lambda$ bounded for which this is the case. If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an infinite dimensional real, complex or quaternionic Hilbert space and $\lambda$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $\pi_\lambda$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $\pi_\lambda$ is norm continuous and our results imply the remarkable result that, for positive energy representations, adding a suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of $\tilde\rho_\lambda(D)$ is $0$ (minimal energy condition). The corresponding pairs $(\lambda,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.



• Conjugacy classes and straight elements in Coxeter groups
  Journal of Algebra 407 (2014), pp. 68-80. pdf | doi | arXiv | abstract

  Abstract:

  Let $W$ be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup $W_0$ of $W$, any two cyclically reduced expressions of conjugate elements of $W_0$ only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in $W_0$. As a byproduct of our methods, we also obtain a characterisation of straight elements of $W$, namely of those elements $w\in W$ for which $\ell(w^n)=n\ell(w)$ for any integer $n$. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight.



• Topological Kac-Moody groups and their subgroups
  Ph.D. thesis (April 2013), Université Catholique de Louvain. pdf | abstract

  Abstract:

  In this thesis, we establish several structure results for topological Kac-Moody groups (minimal or maximal). This thesis may also serve as an introduction to Kac-Moody theory.



• Abstract simplicity of locally compact Kac-Moody groups
  Compositio Math. 150 (2014), pp. 713-728. pdf | doi | arXiv | abstract

  Abstract:

  In this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes an essential use of O. Mathieu’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.



• A fixed point theorem for Lie groups acting on buildings and applications to Kac-Moody theory
  Forum Mathematicum 27 Nr. 1 (2015), pp. 449–466. pdf | doi | arXiv | abstract

  Abstract:

  We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac-Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.



• Open subgroups of locally compact Kac-Moody groups (with Pierre-Emmanuel Caprace)
  Math. Z. 274 Nr. 1-2 (2013), pp. 291–313. pdf | doi | arXiv | abstract

  Abstract:

  Let G be a complete Kac-Moody group over a finite field. It is known that G possesses a BN-pair structure, all of whose parabolic subgroups are open in G. We show that, conversely, every open subgroup of G has finite index in some parabolic subgroup. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.



• Can an anisotropic reductive group admit a Tits system? (with Pierre-Emmanuel Caprace)
  Pure and Appl. Math. Quart. 7 Nr. 3 (2011), pp. 539–558. pdf | doi | arXiv | abstract

  Abstract:

  Seeking for a converse to a well-known theorem by Borel-Tits, we address the question whether the group of rational points $G(k)$ of an anisotropic reductive $k$-group may admit a split spherical BN-pair. We show that if $k$ is a perfect field or a local field, then such a BN-pair must be virtually trivial. We also consider arbitrary compact groups and show that the only abstract BN-pairs they can admit are spherical, and even virtually trivial provided they are split.



• Angle-deformations in Coxeter groups (with Bernhard Mühlherr)
  Algebraic & Geometric Topology 8 (2008), pp. 2175–2208 pdf | doi | arXiv | abstract

  Abstract:

  The isomorphism problem for Coxeter groups has been reduced to its 'reflection preserving version' by B. Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter system $(W,R)$ all Coxeter generating sets S of W which are contained in $R^W$, the set of reflections of $(W,R)$. In this paper, we provide a further reduction: it suffices to determine all Coxeter generating sets $S$ in $R^W$ which are sharp-angled with respect to $R$.