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Journal articles on the topic 'Right-angled group'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Kim, Sang-Hyun, and Thomas Koberda. "The geometry of the curve graph of a right-angled Artin group." International Journal of Algebra and Computation 24, no. 02 (March 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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2

Gutierrez, Mauricio, and 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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3

HAUBOLD, NIKO, MARKUS LOHREY, and CHRISTIAN MATHISSEN. "COMPRESSED DECISION PROBLEMS FOR GRAPH PRODUCTS AND APPLICATIONS TO (OUTER) AUTOMORPHISM GROUPS." International Journal of Algebra and Computation 22, no. 08 (December 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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4

CRISP, JOHN, MICHAH SAGEEV, and MARK SAPIR. "SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS." International Journal of Algebra and Computation 18, no. 03 (May 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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5

Clay, Matt. "When does a right-angled Artin group split over ℤ?" International Journal of Algebra and Computation 24, no. 06 (September 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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6

COSTA, ARMINDO, and MICHAEL FARBER. "TOPOLOGY OF RANDOM RIGHT ANGLED ARTIN GROUPS." Journal of Topology and Analysis 03, no. 01 (March 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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7

Sentinelli, Paolo. "Artin group injection in the Hecke algebra for right-angled groups." Geometriae Dedicata 214, no. 1 (February 22, 2021): 193–210. http://dx.doi.org/10.1007/s10711-021-00611-4.

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8

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

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9

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

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10

Kato, Motoko. "Embeddings of right-angled Artin groups into higher-dimensional Thompson groups." Journal of Algebra and Its Applications 17, no. 08 (July 8, 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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11

Minasyan, Ashot. "On subgroups of right angled Artin groups with few generators." International Journal of Algebra and Computation 25, no. 04 (May 21, 2015): 675–88. http://dx.doi.org/10.1142/s0218196715500150.

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For each d ∈ ℕ, we construct a 3-generated group Hd, which is a subdirect product of free groups, such that the cohomological dimension of Hd is d. Given a group F and a normal subgroup N ⊳ F we prove that any right angled Artin group containing the special HNN-extension of F with respect to N must also contain F/N. We apply this to construct, for every d ∈ ℕ, a 4-generated group Gd, embeddable into a right angled Artin group, such that the cohomological dimension of Gd is 2 but the cohomological dimension of any right angled Artin group, containing Gd, is at least d. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular, we show that for every n ∈ ℕ there exists δ(n) ∈ ℕ such that any n-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the δ(n)-th direct power of the free group of rank 2. As another corollary we derive that any n-generated finitely presented residually free group embeds into the direct product of at most δ(n) limit groups.

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12

Katayama, Takuya. "Embeddability of right-angled Artin groups on the complements of linear forests." Journal of Knot Theory and Its Ramifications 27, no. 01 (January 2018): 1850010. http://dx.doi.org/10.1142/s0218216518500104.

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In this paper, we prove that embeddings of right-angled Artin group [Formula: see text] on the complement of a linear forest into another right-angled Artin group [Formula: see text] can be reduced to full embeddings of the defining graph of [Formula: see text] into the extension graph of the defining graph of [Formula: see text].

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13

BELK, JAMES, COLLIN BLEAK, and FRANCESCO MATUCCI. "Embedding right-angled Artin groups into Brin–Thompson groups." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 2 (April 23, 2019): 225–29. http://dx.doi.org/10.1017/s0305004119000112.

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AbstractWe prove that every finitely-generated right-angled Artin group embeds into some Brin–Thompson group nV. It follows that any virtually special group can be embedded into some nV, a class that includes surface groups, all finitely-generated Coxeter groups, and many one-ended hyperbolic groups.

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14

BRIDSON, MARTIN R. "ON THE RECOGNITION OF RIGHT-ANGLED ARTIN GROUPS." Glasgow Mathematical Journal 62, no. 2 (June 19, 2019): 473–75. http://dx.doi.org/10.1017/s0017089519000235.

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15

Deibel, Angelica. "Random coxeter groups." International Journal of Algebra and Computation 30, no. 06 (August 20, 2020): 1305–21. http://dx.doi.org/10.1142/s0218196720500423.

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Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös–Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about the homology of the nerve of a random Coxeter group and results about when random Coxeter groups are [Formula: see text]-hyperbolic and when they have the FC-type property.

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16

Bregman, Corey, and Neil Fullarton. "Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group." Michigan Mathematical Journal 66, no. 3 (August 2017): 569–80. http://dx.doi.org/10.1307/mmj/1496995334.

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17

Wade, Richard. "The lower central series of a right-angled Artin group." L’Enseignement Mathématique 61, no. 3 (2015): 343–71. http://dx.doi.org/10.4171/lem/61-3/4-4.

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18

Veryovkin, Ya A. "The Associated Lie Algebra of a Right-Angled Coxeter Group." Proceedings of the Steklov Institute of Mathematics 305, no. 1 (May 2019): 53–62. http://dx.doi.org/10.1134/s0081543819030040.

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19

Panov, Taras, and Yakov Veryovkin. "On the commutator subgroup of a right-angled Artin group." Journal of Algebra 521 (March 2019): 284–98. http://dx.doi.org/10.1016/j.jalgebra.2018.11.022.

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20

Baik, Hyungryul, Sang-hyun Kim, and Thomas Koberda. "Right-angled Artin groups in the C ∞ diffeomorphism group of the real line." Israel Journal of Mathematics 213, no. 1 (April 15, 2016): 175–82. http://dx.doi.org/10.1007/s11856-016-1307-8.

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21

Day, Matthew B. "Symplectic structures on right-angled Artin groups: Between the mapping class group and the symplectic group." Geometry & Topology 13, no. 2 (January 8, 2009): 857–99. http://dx.doi.org/10.2140/gt.2009.13.857.

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22

Corwin, Nathan, and Kathryn Haymaker. "The graph structure of graph groups that are subgroups of Thompson’s group V." International Journal of Algebra and Computation 26, no. 08 (December 2016): 1497–501. http://dx.doi.org/10.1142/s021819671650065x.

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We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson’s group [Formula: see text]. It was shown by Bleak and Salazar-Díaz that [Formula: see text] was an obstruction. We show that this is the only obstruction. This is shown by proving a graph theory result giving an alternate description of simple graphs without an appropriate induced subgraph.

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23

Hambleton, Ian, and Alyson Hildum. "Topological 4-manifolds with right-angled Artin fundamental groups." Journal of Topology and Analysis 11, no. 04 (December 2019): 777–821. http://dx.doi.org/10.1142/s1793525319500328.

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We classify closed, spin[Formula: see text], topological [Formula: see text]-manifolds with fundamental group [Formula: see text] of cohomological dimension [Formula: see text] (up to [Formula: see text]-cobordism), after stabilization by connected sum with at most [Formula: see text] copies of [Formula: see text]. In general, we must also assume that [Formula: see text] satisfies certain [Formula: see text]-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of [Formula: see text]-manifolds and all right-angled Artin groupswhose defining graphs have no [Formula: see text]-cliques.

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24

De Medts, Tom, and Ana C. Silva. "Open subgroups of the automorphism group of a right-angled building." Geometriae Dedicata 203, no. 1 (January 14, 2019): 1–23. http://dx.doi.org/10.1007/s10711-019-00423-7.

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25

Katayama, Takuya. "Right-angled Artin groups and full subgraphs of graphs." Journal of Knot Theory and Its Ramifications 26, no. 10 (September 2017): 1750059. http://dx.doi.org/10.1142/s0218216517500596.

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For a finite graph [Formula: see text], let [Formula: see text] be the right-angled Artin group defined by the complement graph of [Formula: see text]. We show that, for any linear forest [Formula: see text] and any finite graph [Formula: see text], [Formula: see text] can be embedded into [Formula: see text] if and only if [Formula: see text] can be realized as a full subgraph of [Formula: see text]. We also prove that if we drop the assumption that [Formula: see text] is a linear forest, then the above assertion does not hold, namely, for any finite graph [Formula: see text], which is not a linear forest, there exists a finite graph [Formula: see text] such that [Formula: see text] can be embedded into [Formula: see text], though [Formula: see text] cannot be embedded into [Formula: see text] as a full subgraph.

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26

Clay, Matt, Johanna Mangahas, and Dan Margalit. "Right-angled Artin groups as normal subgroups of mapping class groups." Compositio Mathematica 157, no. 8 (July 27, 2021): 1807–52. http://dx.doi.org/10.1112/s0010437x21007417.

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We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

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27

DAY, MATTHEW B. "ON SOLVABLE SUBGROUPS OF AUTOMORPHISM GROUPS OF RIGHT-ANGLED ARTIN GROUPS." International Journal of Algebra and Computation 21, no. 01n02 (February 2011): 61–70. http://dx.doi.org/10.1142/s021819671100608x.

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For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.

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28

PY, PIERRE. "Coxeter groups and Kähler groups." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 3 (September 2, 2013): 557–66. http://dx.doi.org/10.1017/s0305004113000534.

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AbstractWe study homomorphisms from Kähler groups to Coxeter groups. As an application, we prove that a cocompact complex hyperbolic lattice (in complex dimension at least 2) does not embed into a Coxeter group or a right-angled Artin group. This is in contrast with the case of real hyperbolic lattices.

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29

Lee, Eon-Kyung, and Sang-Jin Lee. "Embeddability of right-angled Artin groups on complements of trees." International Journal of Algebra and Computation 28, no. 03 (May 2018): 381–94. http://dx.doi.org/10.1142/s0218196718500182.

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For a finite simplicial graph [Formula: see text], let [Formula: see text] denote the right-angled Artin group on [Formula: see text]. Recently, Kim and Koberda introduced the extension graph [Formula: see text] for [Formula: see text], and established the Extension Graph Theorem: for finite simplicial graphs [Formula: see text] and [Formula: see text], if [Formula: see text] embeds into [Formula: see text] as an induced subgraph then [Formula: see text] embeds into [Formula: see text]. In this paper, we show that the converse of this theorem does not hold for the case [Formula: see text] is the complement of a tree and for the case [Formula: see text] is the complement of a path graph.

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30

Bell, Robert W. "Combinatorial Methods for Detecting Surface Subgroups in Right-Angled Artin Groups." ISRN Algebra 2011 (August 23, 2011): 1–6. http://dx.doi.org/10.5402/2011/102029.

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We give a short proof of the following theorem of Sang-hyun Kim: if is a right-angled Artin group with defining graph , then contains a hyperbolic surface subgroup if contains an induced subgraph for some , where denotes the complement graph of an -cycle. Furthermore, we give a new proof of Kim's cocontraction theorem.

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31

Abbott, Carolyn, Jason Behrstock, and Matthew Durham. "Largest acylindrical actions and Stability in hierarchically hyperbolic groups." Transactions of the American Mathematical Society, Series B 8, no. 3 (February 16, 2021): 66–104. http://dx.doi.org/10.1090/btran/50.

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We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3 3 –manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an “almost hierarchically hyperbolic space” is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.

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Félix, Yves, and Steve Halperin. "The depth of a Riemann surface and of a right-angled Artin group." Journal of Homotopy and Related Structures 15, no. 1 (November 12, 2019): 223–48. http://dx.doi.org/10.1007/s40062-019-00250-3.

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Fullarton, Neil J. "On the number of outer automorphisms of the automorphism group of a right-angled Artin group." Mathematical Research Letters 23, no. 1 (2016): 145–62. http://dx.doi.org/10.4310/mrl.2016.v23.n1.a8.

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34

Ciobanu, Laura, Charles Garnet Cox, and Armando Martino. "The Conjugacy Ratio of Groups." Proceedings of the Edinburgh Mathematical Society 62, no. 3 (February 22, 2019): 895–911. http://dx.doi.org/10.1017/s0013091518000573.

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AbstractIn this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group.

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Kharlampovich, Olga, and Alexei Myasnikov. "Equations in Algebras." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1517–33. http://dx.doi.org/10.1142/s0218196718400064.

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We show that the Diophantine problem (decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right-angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc.

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Naik, Tushar K., Neha Nanda, and Mahender Singh. "Some remarks on twin groups." Journal of Knot Theory and Its Ramifications 29, no. 10 (August 19, 2020): 2042006. http://dx.doi.org/10.1142/s0218216520420067.

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The twin group [Formula: see text] is a right angled Coxeter group generated by [Formula: see text] involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups [Formula: see text] have [Formula: see text]-property and are not co-Hopfian for [Formula: see text].

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Kambites, Mark. "On commuting elements and embeddings of graph groups and monoids." Proceedings of the Edinburgh Mathematical Society 52, no. 1 (February 2009): 155–70. http://dx.doi.org/10.1017/s0013091507000119.

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AbstractWe study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.

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Koban, Nic, and Adam Piggott. "The Bieri–Neumann–Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group." Illinois Journal of Mathematics 58, no. 1 (2014): 27–41. http://dx.doi.org/10.1215/ijm/1427897167.

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ESTRADA, B., and E. MARTÍNEZ. "COORDINATES FOR THE TEICHMÜLLER SPACE OF PLANAR SURFACE N.E.C. GROUPS." International Journal of Mathematics 14, no. 10 (December 2003): 1037–52. http://dx.doi.org/10.1142/s0129167x03002137.

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It is proved that every planar surface NEC group Γ admits a fundamental region, called a normal region, which is a hyperbolic right-angled polygon with several pairs of identified sides according to a pattern. Conversely, each such a polygon can be taken as a fundamental region of a planar surface NEC group. By means of these regions the Teichmüller space of Γ is studied, obtaining a parametrization by means of congruence classes of marked polygons. The parameters are certain lengths in these polygons, and from them explicit matrices of the generators of the associated group are obtained.

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CHARNEY, RUTH. "THE TITS CONJECTURE FOR LOCALLY REDUCIBLE ARTIN GROUPS." International Journal of Algebra and Computation 10, no. 06 (December 2000): 783–97. http://dx.doi.org/10.1142/s0218196700000479.

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Given an Artin system (A,S), a conjecture of Tits states that the subgroup A(2) of A generated by the squares of the generators in S is subject only to the obvious commutator relations between generators. In particular, A(2) is a right-angled Artin group. We prove this conjecture for a class of infinite type Artin groups, called locally reducible Artin groups, for which the associated Deligne complex has a CAT(0) geometry. We also prove that for any special subgroup AT of A, A(2)∩AT=(AT)(2).

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ARZHANTSEVA, GOULNARA N., and CHRISTOPHER H. CASHEN. "Cogrowth for group actions with strongly contracting elements." Ergodic Theory and Dynamical Systems 40, no. 7 (December 4, 2018): 1738–54. http://dx.doi.org/10.1017/etds.2018.123.

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Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

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Capdeboscq, Inna, and Anne Thomas. "Co-compact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups." Mathematical Research Letters 20, no. 2 (2013): 339–58. http://dx.doi.org/10.4310/mrl.2013.v20.n2.a10.

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43

Veryovkin, Ya A. "Graded Components of the Lie Algebra Associated with the Lower Central Series of a Right-Angled Coxeter Group." Proceedings of the Steklov Institute of Mathematics 318, no. 1 (September 2022): 26–37. http://dx.doi.org/10.1134/s0081543822040034.

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44

Nashold, Blaine S., Amr O. El-Naggar, Janice Ovelmen-Levitt, and Muwaffak Abdul-Hak. "A new design of radiofrequency lesion electrodes for use in the caudalis nucleus DREZ operation." Journal of Neurosurgery 80, no. 6 (June 1994): 1116–20. http://dx.doi.org/10.3171/jns.1994.80.6.1116.

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✓ Two new right-angled electrodes have been designed for use at the dorsal root entry zone (DREZ) of the caudalis nucleus to provide relief of chronic facial pain. The electrode design was based on an anatomical study of the human caudalis nucleus at the cervicomedullary junction. Previously, caudalis nucleus DREZ operations were often followed by ipsilateral ataxia, usually in the arm. The new electrodes have significantly reduced this complication. A group of 21 patients with varied types of chronic facial pain have been treated, with pain relief in 70%.

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45

González, Jesús, Bárbara Gutiérrez, and Hugo Mas. "Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups." Journal of Topology and Analysis 11, no. 02 (June 2019): 371–86. http://dx.doi.org/10.1142/s1793525319500171.

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The clique number of a random graph in the Erdös–Rényi model [Formula: see text] yields a random variable which takes values asymptotically almost surely (as [Formula: see text]) within one of an explicit logarithmic function [Formula: see text]. We show that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint cliques with [Formula: see text] vertices. Such a result is motivated by, and applied to, the multi-tasking version of Farber’s topological model to study the motion planning problem in robotics. Indeed, we study the behavior of all the higher topological complexities of Eilenberg–MacLane spaces of type [Formula: see text], where [Formula: see text] is a random right angled Artin group.

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46

DUNCAN, ANDREW J., and VLADIMIR N. REMESLENNIKOV. "AUTOMORPHISMS OF PARTIALLY COMMUTATIVE GROUPS II: COMBINATORIAL SUBGROUPS." International Journal of Algebra and Computation 22, no. 07 (November 2012): 1250074. http://dx.doi.org/10.1142/s0218196712500749.

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We define several "standard" subgroups of the automorphism group Aut (G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut (G). If C is the commutation graph of G, we show how Aut (G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decomposition of Aut (G) into a subgroup of locally conjugating automorphisms by the subgroup stabilizing a certain lattice of "admissible subsets" of the vertices of C. We then characterize those graphs for which Aut (G) is a product (not necessarily semi-direct) of two such subgroups.

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47

Schesler, Eduard. "The relative exponential growth rate of subgroups of acylindrically hyperbolic groups." Journal of Group Theory 25, no. 2 (October 5, 2021): 293–326. http://dx.doi.org/10.1515/jgth-2020-0180.

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Abstract We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

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48

Schesler, Eduard. "The relative exponential growth rate of subgroups of acylindrically hyperbolic groups." Journal of Group Theory 25, no. 2 (October 5, 2021): 293–326. http://dx.doi.org/10.1515/jgth-2020-0180.

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Abstract We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

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49

Naik, Tushar Kanta, Neha Nanda, and Mahender Singh. "Conjugacy classes and automorphisms of twin groups." Forum Mathematicum 32, no. 5 (September 1, 2020): 1095–108. http://dx.doi.org/10.1515/forum-2019-0321.

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AbstractThe twin group {T_{n}} is a right-angled Coxeter group generated by {n-1} involutions, and the pure twin group {\mathrm{PT}_{n}} is the kernel of the natural surjection from {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in {T_{n}}, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in {T_{n}}. We give a new proof of the structure of {\operatorname{Aut}(T_{n})} for {n\geq 3}, and show that {T_{n}} is isomorphic to a subgroup of {\operatorname{Aut}(\mathrm{PT}_{n})} for {n\geq 4}. Finally, we construct a representation of {T_{n}} to {\operatorname{Aut}(F_{n})} for {n\geq 2}.

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50

Gray, Robert D. "Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups." Inventiones mathematicae 219, no. 3 (September 9, 2019): 987–1008. http://dx.doi.org/10.1007/s00222-019-00920-2.

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Abstract We prove the following results: (1) There is a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$Inv⟨A|w=1⟩ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg (J Algebra 320(2):728–755, 2008), we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$Inv⟨A|w=1⟩ with undecidable word problem. Furthermore, this method allows the construction of an E-unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis et al. (in: Semigroups and their applications, Reidel, Dordrecht, pp. 99–110, 1987).

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