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Littérature scientifique sur le sujet « Right-angled group »

Auteur : Grafiati
Publié le 9 mars 2023
Mis à jour le 10 mars 2023

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Articles de revues sur le sujet "Right-angled group"

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Kim, Sang-Hyun, et Thomas Koberda. « The geometry of the curve graph of a right-angled Artin group ». International Journal of Algebra and Computation 24, no 02 (mars 2014) : 121–69. http://dx.doi.org/10.1142/s021819671450009x.

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We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph, respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result, we are able to develop a Nielsen–Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.
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Gutierrez, Mauricio, et Anton Kaul. « Automorphisms of Right-Angled Coxeter Groups ». International Journal of Mathematics and Mathematical Sciences 2008 (2008) : 1–10. http://dx.doi.org/10.1155/2008/976390.

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If is a right-angled Coxeter system, then is a semidirect product of the group of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, is a semidirect product of by the quotient . We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence and consequently, all group extensions are trivial.
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HAUBOLD, NIKO, MARKUS LOHREY et CHRISTIAN MATHISSEN. « COMPRESSED DECISION PROBLEMS FOR GRAPH PRODUCTS AND APPLICATIONS TO (OUTER) AUTOMORPHISM GROUPS ». International Journal of Algebra and Computation 22, no 08 (décembre 2012) : 1240007. http://dx.doi.org/10.1142/s0218196712400073.

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It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a restricted variant of the simultaneous compressed conjugacy problem is polynomial time Turing-reducible to the same problem for the vertex groups. A direct corollary of this result is that the word problem for the outer automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Finally, it is shown that the compressed variant of the ordinary conjugacy problem can be solved in polynomial time for right-angled Artin groups.
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CRISP, JOHN, MICHAH SAGEEV et MARK SAPIR. « SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS ». International Journal of Algebra and Computation 18, no 03 (mai 2008) : 443–91. http://dx.doi.org/10.1142/s0218196708004536.

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We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight "forbidden" graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right-angled Artin group to contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs P2(6), the right-angled Artin group A(P2(6)) is a subgroup of a (right-angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).
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Clay, Matt. « When does a right-angled Artin group split over ℤ ? » International Journal of Algebra and Computation 24, no 06 (septembre 2014) : 815–25. http://dx.doi.org/10.1142/s0218196714500350.

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We show that a right-angled Artin group, defined by a graph Γ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if Γ is biconnected. Further, we compute JSJ-decompositions of 1-ended right-angled Artin groups over infinite cyclic subgroups.
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COSTA, ARMINDO, et MICHAEL FARBER. « TOPOLOGY OF RANDOM RIGHT ANGLED ARTIN GROUPS ». Journal of Topology and Analysis 03, no 01 (mars 2011) : 69–87. http://dx.doi.org/10.1142/s1793525311000490.

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In this paper, we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting complexity of motion planning algorithms in robotics. We show that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values, when n → ∞. We use a result of Cohen and Pruidze which expresses the topological complexity of right angled Artin groups in combinatorial terms. Our proof deals with the existence of bi-cliques in random graphs.
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Sentinelli, Paolo. « Artin group injection in the Hecke algebra for right-angled groups ». Geometriae Dedicata 214, no 1 (22 février 2021) : 193–210. http://dx.doi.org/10.1007/s10711-021-00611-4.

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Jensen, C., et J. Meier. « The Cohomology of Right-Angled Artin Groups with Group Ring Coefficients ». Bulletin of the London Mathematical Society 37, no 5 (octobre 2005) : 711–18. http://dx.doi.org/10.1112/s0024609305004571.

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Paolini, Gianluca, et Saharon Shelah. « No Uncountable Polish Group Can be a Right-Angled Artin Group ». Axioms 6, no 4 (11 mai 2017) : 13. http://dx.doi.org/10.3390/axioms6020013.

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Kato, Motoko. « Embeddings of right-angled Artin groups into higher-dimensional Thompson groups ». Journal of Algebra and Its Applications 17, no 08 (8 juillet 2018) : 1850159. http://dx.doi.org/10.1142/s0219498818501591.

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In this paper, we show that every right-angled Artin group embeds into the [Formula: see text]-dimensional Thompson group [Formula: see text] for some [Formula: see text], by constructing a set of embeddings. This is an improvement of the previous result of Belk, Bleak and Matucci, which gives a different set of embeddings from right-angled Artin groups into higher-dimensional Thompson groups. Compared to their construction, the dimension [Formula: see text] of the target group is always smaller with our construction.
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Thèses sur le sujet "Right-angled group"

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Toinet, Emmanuel. « Automorphisms of right-angled Artin groups ». Thesis, Dijon, 2012. http://www.theses.fr/2012DIJOS003.

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Cette thèse a pour objet l’étude des automorphismes des groupes d’Artin à angles droits. Etant donné un graphe simple fini G, le groupe d’Artin à angles droits GG associé à G est le groupe défini par la présentation dont les générateurs sont les sommets de G, et dont les relateurs sont les commutateurs [v,w], où {v,w} est une paire de sommets adjacents. Le premier chapitre est conçu comme une introduction générale à la théorie des groupes d’Artin à angles droits et de leurs automorphismes. Dans un deuxième chapitre, on démontre que tout sous-groupe sous-normal d’indice une puissance de p d’un groupe d’Artin à angles droits est résiduellement p-séparable. Comme application de ce résultat, on montre que tout groupe d’Artin à angles droits est résiduellement séparable dans la classe des groupes nilpotents sans torsion. Une autre application de ce résultat est que le groupe des automorphismes extérieurs d’un groupe d’Artin à angles droits est virtuellement résiduellement p-fini. On montre également que le groupe de Torelli d’un groupe d’Artin à angles droits est résiduellement nilpotent sans torsion, et, par suite, résiduellement p-fini et bi-ordonnable. Dans un troisième chapitre, on établit une présentation du sous-groupe Conj(GG) deAut(GG) formé des automorphismes qui envoient chaque générateur sur un conjugué de lui-même
The purpose of this thesis is to study the automorphisms of right-angled Artin groups. Given a finite simplicial graph G, the right-angled Artin group GG associated to G is the group defined by the presentation whose generators are the vertices of G, and whose relators are commuta-tors of pairs of adjacent vertices. The first chapter is intended as a general introduction to the theory of right-angled Artin groups and their automor-phisms. In a second chapter, we prove that every subnormal subgroup ofp-power index in a right-angled Artin group is conjugacy p-separable. As an application, we prove that every right-angled Artin group is conjugacy separable in the class of torsion-free nilpotent groups. As another applica-tion, we prove that the outer automorphism group of a right-angled Artin group is virtually residually p-finite. We also prove that the Torelli group ofa right-angled Artin group is residually torsion-free nilpotent, hence residu-ally p-finite and bi-orderable. In a third chapter, we give a presentation of the subgroup Conj(GG) of Aut(GG) consisting of the automorphisms thats end each generator to a conjugate of itself
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Wade, Richard D. « Symmetries of free and right-angled Artin groups ». Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:b856e2b5-3689-472b-95c1-71b5748affc9.

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The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free group and if we take all possible relations we have a free abelian group. We show that if no finite index subgroup of a group $G$ contains a normal subgroup that maps onto $mathbb{Z}$, then every homomorphism from $G$ to the outer automorphism group of a free group has finite image. The above criterion is satisfied by SL$_m(mathbb{Z})$ for $m geq 3$ and, more generally, all irreducible lattices in higher-rank, semisimple Lie groups with finite centre. Given a right-angled Artin group $A_Gamma$ we find an integer $n$, which may be easily read off from the presentation of $A_G$, such that if $m geq 3$ then SL$_m(mathbb{Z})$ is a subgroup of the outer automorphism group of $A_Gamma$ if and only if $m leq n$. More generally, we find criteria to prevent a group from having a homomorphism to the outer automorphism group of $A_Gamma$ with infinite image, and apply this to a large number of irreducible lattices as above. We study the subgroup $IA(A_Gamma)$ of $Aut(A_Gamma)$ that acts trivially on the abelianisation of $A_Gamma$. We show that $IA(A_Gamma)$ is residually torsion-free nilpotent and describe its abelianisation. This is complemented by a survey of previous results concerning the lower central series of $A_Gamma$. One of the commonly used generating sets of $Aut(F_n)$ is the set of Whitehead automorphisms. We describe a geometric method for decomposing an element of $Aut(F_n)$ as a product of Whitehead automorphisms via Stallings' folds. We finish with a brief discussion of the action of $Out(F_n)$ on Culler and Vogtmann's Outer Space. In particular we describe translation lengths of elements with regards to the `non-symmetric Lipschitz metric' on Outer Space.
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Bounds, Jordan. « On the quasi-isometric rigidity of a class of right-angled Coxeter groups ». Bowling Green State University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1561561078356503.

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FONIQI, ISLAM. « Results on Artin and twisted Artin groups ​ ». Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/374264.

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Questa tesi consiste in tre capitoli principali, e tutti si evolvono intorno ai gruppi di Artin. Dimostrare risultati per tutti i gruppi di Artin è una sfida seria, quindi di solito ci si concentra su particolari sottoclassi. Tra le sottofamiglie più conosciute dei gruppi di Artin c'è la famiglia dei gruppi Artin ad angolo retto (RAAGs in breve). Si possono definire usando i grafici simpliciali, che determinano il gruppo fino all'isomorfismo. Sono anche interessanti perché ci sono una varietà di metodi per studiarli, provenienti da diversi punti di vista, come la geometria, l'algebra e la combinatoria. Questo ha portato alla comprensione di molti problemi dei RAAG, come il problema delle parole, la crescita sferica, le intersezioni di sottogruppi parabolici, ecc. Nel Capitolo 2 ci concentriamo sulla crescita geodetica dei RAAG, su grafi link-regolari, ed estendiamo un risultato in quella direzione, fornendo una formula della crescita su grafi link-regolari senza tetraedri. Nel capitolo 3 lavoriamo con gruppi leggermente diversi, la classe dei gruppi Artin contorti ad angolo retto (tRAAGs in breve). Sono definiti usando grafi misti, che sono grafi semplici in cui i bordi possono essere diretti. Troviamo una forma normale per presentare gli elementi in un tRAAG. Se dimentichiamo le direzioni dei bordi, otteniamo un grafo non diretto sottostante, che chiamiamo grafo ingenuo. Sul grafo ingenuo, che è semplice, si può definire un RAAG, che corrisponde naturalmente al nostro tRAAG. Discuteremo alcune somiglianze e differenze algebriche e geometriche tra i tRAAG e i RAAG. Usando la forma normale siamo in grado di concludere che la crescita sferica e geodetica di un tRAAG concorda con la rispettiva crescita del RAAG sottostante. Il capitolo 4 ha un tema diverso, e consiste nello studio dei sottogruppi parabolici nei gruppi pari di Artin. Il lavoro è motivato dai risultati corrispondenti nei RAAG, e generalizziamo alcuni di questi risultati a certe sottoclassi di gruppi pari di Artin. ​
This thesis consists of three main chapters, and they all evolve around Artin groups. Proving results for all Artin groups is a serious challenge, so one usually focuses on particular subclasses. Among the most well-understood subfamilies of Artin groups is the family of right-angled Artin groups (RAAGs shortly). One can define them using simplicial graphs, which determine the group up to isomorphism. They are also interesting as there are a variety of methods for studying them, coming from different viewpoints, such as geometry, algebra, and combinatorics. This has resulted in the understanding of many problems in RAAGs, like the word problem, the spherical growth, intersections of parabolic subgroups, etc. In Chapter 2 we focus on the geodesic growth of RAAGs, over link-regular graphs, and we extend a result in that direction, by providing a formula of the growth over link-regular graphs without tetrahedrons. In Chapter 3 we work with slightly different groups, the class of twisted right-angled Artin groups (tRAAGs shortly). They are defined using mixed graphs, which are simplicial graphs where edges are allowed to be directed edges. We find a normal form for presenting the elements in a tRAAG. If we forget about the directions of edges, we obtain an underlying undirected graph, which we call the naïve graph. Over the naïve graph, which is simplicial, one can define a RAAG, which corresponds naturally to our tRAAG. We will discuss some algebraic and geometric similarities and differences between tRAAGs and RAAGs. Using the normal form we are able to conclude that the spherical and geodesic growth of a tRAAG agrees with the respective growth of the underlying RAAG. Chapter 4 has a different theme, and it consists of the study of parabolic subgroups in even Artin groups. The work is motivated by the corresponding results in RAAGs, and we generalize some of these results to certain subclasses of even Artin groups. ​
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Fullarton, Neil James. « Palindromic automorphisms of free groups and rigidity of automorphism groups of right-angled Artin groups ». Thesis, University of Glasgow, 2014. http://theses.gla.ac.uk/5323/.

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Let F_n denote the free group of rank n with free basis X. The palindromic automorphism group PiA_n of F_n consists of automorphisms taking each member of X to a palindrome: that is, a word on X that reads the same backwards as forwards. We obtain finite generating sets for certain stabiliser subgroups of PiA_n. We use these generating sets to find an infinite generating set for the so-called palindromic Torelli group PI_n, the subgroup of PiA_n consisting of palindromic automorphisms inducing the identity on the abelianisation of F_n. Two crucial tools for finding this generating set are a new simplicial complex, the so-called complex of partial pi-bases, on which PiA_n acts, and a Birman exact sequence for PiA_n, which allows us to induct on n. We also obtain a rigidity result for automorphism groups of right-angled Artin groups. Let G be a finite simplicial graph, defining the right-angled Artin group A_G. We show that as A_G ranges over all right-angled Artin groups, the order of Out(Aut(A_G)) does not have a uniform upper bound. This is in contrast with extremal cases when A_G is free or free abelian: in these cases, |Out(Aut(A_G))| < 5. We prove that no uniform upper bound exists in general by placing constraints on the graph G that yield tractable decompositions of Aut(A_G). These decompositions allow us to construct explicit members of Out(Aut(A_G)).
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Karrer, Annette [Verfasser], et P. [Akademischer Betreuer] Schwer. « Contracting boundaries of amalgamated free products of CAT(0) groups with applications for right-angled Coxeter groups / Annette Karrer ; Betreuer : P. Schwer ». Karlsruhe : KIT-Bibliothek, 2021. http://d-nb.info/1227450982/34.

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Girão, Darlan Rabelo. « Rank gradient in co-final towers of certain Kleinian groups ». Thesis, 2011. http://hdl.handle.net/2152/ETD-UT-2011-12-4673.

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This dissertation provides the first known examples of finite co-volume Kleinian groups which have co- final towers of finite index subgroups with positive rank gradient. We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has fi nite index in the reflection group of a right-angled ideal polyhedron in H^3 then it has a co-fi nal tower of fi nite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co- final towers of covers with zero rank gradient. We also prove that the reflection groups of compact right-angled hyperbolic polyhedra satisfying mild conditions have co-fi nal towers of fi nite sheeted covers with positive rank gradient.
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Livres sur le sujet "Right-angled group"

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From riches to raags : 3-manifolds, right-angled artin groups, and cubical geometry. Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, Rhode Island with support from the National Science Foundation, 2012.

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Chapitres de livres sur le sujet "Right-angled group"

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Koberda, Thomas. « Geometry and Combinatorics via Right-Angled Artin Groups ». Dans In the Tradition of Thurston II, 475–518. Cham : Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-97560-9_15.

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Bell, Robert W., et Matt Clay. « Right-Angled Artin Groups ». Dans Office Hours with a Geometric Group Theorist. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691158662.003.0014.

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This chapter deals with right-angled Artin groups, a broad spectrum of groups that includes free groups on one end, free abelian groups on the other end, and many other interesting groups in between. A right-angled Artin group is a group G(Γ‎) defined in terms of a graph Γ‎. Right-angled Artin groups have taken a central role in geometric group theory, mainly due to their involvement in the solution to one of the main open questions in the topology of 3-manifolds. The chapter first considers right-angled Artin groups as subgroups and how they relate to other classes of groups before exploring subgroups of right-angled Artin groups and the word problem for right-angled Artin groups. The discussion includes exercises and research projects.
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« 14. Right-Angled Artin Groups ». Dans Office Hours with a Geometric Group Theorist, 291–309. Princeton University Press, 2017. http://dx.doi.org/10.1515/9781400885398-016.

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Otuma, Nick Vincent. « Mismatch between Spoken Language and Visual Representation of Mathematical Concepts ». Dans Building on the Past to Prepare for the Future, Proceedings of the 16th International Conference of The Mathematics Education for the Future Project, King's College,Cambridge, Aug 8-13, 2022, 384–88. WTM-Verlag, 2022. http://dx.doi.org/10.37626/ga9783959872188.0.073.

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This paper examines secondary students’ mismatch in meaning between spoken language and visual representation of mathematical concept of a rightangled triangle. Forty-eight students, age 16-17years participated in the case study. Students were asked to select plane figures that matched the descriptions given on each questionnaire item. In group interview, participants were asked to give properties of selected plane figures and draw a diagram representing the same plane figures. The results of this research suggested that many students had similar imperfect conception of a right-angled triangle. Keywords: Mathematical language, conceptual understanding.
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« Cubulating malnormal graphs of cubulated groups ». Dans From Riches to Raags : 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 69–76. Providence, Rhode Island : American Mathematical Society, 2012. http://dx.doi.org/10.1090/cbms/117/08.

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« Hyperbolic groups with a quasiconvex hierachy ». Dans From Riches to Raags : 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 121–24. Providence, Rhode Island : American Mathematical Society, 2012. http://dx.doi.org/10.1090/cbms/117/14.

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« Overview ». Dans From Riches to Raags : 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 1–5. Providence, Rhode Island : American Mathematical Society, 2012. http://dx.doi.org/10.1090/cbms/117/01.

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« Nonpositively curved cube complexes ». Dans From Riches to Raags : 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 7–14. Providence, Rhode Island : American Mathematical Society, 2012. http://dx.doi.org/10.1090/cbms/117/02.

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« Cubical disk diagrams, hyperplanes, and convexity ». Dans From Riches to Raags : 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 15–30. Providence, Rhode Island : American Mathematical Society, 2012. http://dx.doi.org/10.1090/cbms/117/03.

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« Special cube complexes ». Dans From Riches to Raags : 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 31–42. Providence, Rhode Island : American Mathematical Society, 2012. http://dx.doi.org/10.1090/cbms/117/04.

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Actes de conférences sur le sujet "Right-angled group"

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Belluco, Rosana Zabulon Feijó, Flávio Lúcio Vasconcelos, Paulo Eduardo Silva Belluco, Júllia Eduarda Feijó Belluco et Carmelia Matos Santiago Reis. « NIPPLE MINIMUM PAGET DISEASE : A CASE REPORT ». Dans XXIV Congresso Brasileiro de Mastologia. Mastology, 2022. http://dx.doi.org/10.29289/259453942022v32s1059.

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Paget’s disease (PD) of the nipple is a rare cancer that affects the nipple and areola and accounts for between 0.4% and 5% of breast cancers. It was first described in 1877 by the English physician Sir James Paget. It affects women between 60 and 70 years of age and very rarely affects men. In PD, the skin on the nipple and areola becomes thicker. Clinical presentations are usually erythema, desquamation, or eczematous changes in the nipple, features that can progressively progress to erosion, overt destruction, and ulceration of the papilla. Bloody papillary discharge, itching, nipple retraction, and/or a palpable mass may be associated. Cancer cells, called Paget cells, are malignant, large, with clear, abundant cytoplasm and nuclei with prominent nucleoli. Like glandular cells, they appear either as isolated cell in the epidermal tissue or as groups of cells. Most women diagnosed with PD also have ductal adenocarcinoma, either in situ or invasive. The prevalence is 67–100% of cases, which gives a worse prognosis to the patient. Patients with Paget-associated invasive breast disease have lower hormone receptor expression, greater lymph node involvement, and higher human epidermal growth factor receptor type 2 (HER2) expression. An 82-year-old woman sought the mastology outpatient clinic for a follow-up of carcinoma in situ in the right breast 2 years ago, having been submitted to quadrantectomy and hormone therapy with tamoxifen, with no signs of recurrence. She complained of an exudative pruritic lesion on the left nipple that had started 6 months ago. She reported that the lesion started with itching and burning, associated with a spontaneous discharge of serous secretion from the itchy surface of the breast, which improved with the use of “talcum powder.” On physical examination, the presence of a discrete reddened area with a diameter of 3 mm, eczematous, with bloody areas interspersed with serous secretion was observed on the left nipple. Areola lesions and palpable nodules in the left breast were absent. She underwent mammography, which showed symmetrical breasts with fat-replaced parenchyma, absence of nodules, presence of isolated calcifications, and grouping in the superior lateral region of the left breast, categorized as BIRADS II. On ultrasound, a nodule with angled edges, measuring 5×4 mm in the superomedial quadrant of the left breast, which showed nodular enhancement and persistent kinetic curve on magnetic resonance imaging of the breasts. The histopathological study diagnosed moderately differentiated left breast ductal carcinoma, associated with a high-grade solid intraductal carcinoma and PD of the nipple, without the involvement of the areola. Immunohistochemistry revealed the absence of estrogen and progesterone hormone receptors and HER-2 overexpression in both histological types. She underwent mastectomy with sentinel lymph node biopsy that was free of neoplasia. Oncological follow-up with no signs of recurrence. PD, if left untreated, extends to the areola and other regions of the breast. Therefore, clinical suspicion from the first physical examination allows an early diagnosis of extreme importance, which improves the prognosis and allows less aggressive treatments.
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