Sarah Rees
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View article: Rewriting in Artin groups without A_3 or B_3 subdiagrams
Rewriting in Artin groups without A_3 or B_3 subdiagrams Open
We prove that the word problem in an Artin group G based on a diagram without A_3 or B_3 subdiagrams can be solved using a system of length preserving rewrite rules which, together with free reduction, can be used to reduce any word over t…
View article: Groups with ET0L co-word problem
Groups with ET0L co-word problem Open
We study groups whose co-word problems are ET0L languages, which we call coET0L groups, using an automaton based model due to van Leeuwen, and recently studied by Bishop and Elder. In particular we prove a number of closure results for the…
View article: The compressed conjugacy problem in relatively hyperbolic groups
The compressed conjugacy problem in relatively hyperbolic groups Open
We prove that the compressed conjugacy problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
View article: The Artin monoid Cayley graph
The Artin monoid Cayley graph Open
In this paper, we investigate properties of the Artin monoid Cayley graph. This is the Cayley graph of an Artin group A_{\Gamma} with respect to the (infinite) generating set given by the associated Artin monoid A^+_{\Gamma} . In a previou…
View article: Unsafe in a Safe Country: Evaluating the Support and Reintegration Challenges for Albanian Trafficking Victims within the Context of the UK’s “Safe Country” Designation
Unsafe in a Safe Country: Evaluating the Support and Reintegration Challenges for Albanian Trafficking Victims within the Context of the UK’s “Safe Country” Designation Open
This research delves into the multifaceted challenges of human trafficking and domestic violence in Albania, set against the backdrop of its designation as a "safe country of origin" by the United Kingdom. It critically examines the implic…
View article: Artin groups of type (2,3,n)
Artin groups of type (2,3,n) Open
Our main theorem is that the word problem in the Artin group G = for n >= 5 can be solved using a system R of length preserving rewrite rules that, together with free reduction, can be used to reduce any word over {a,b,c} to a geodesic wor…
View article: Interval groups related to finite Coxeter groups Part II
Interval groups related to finite Coxeter groups Part II Open
We provide a complete description of the presentations of the interval groups related to quasi‐Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associ…
View article: Interval groups related to finite Coxeter groups I
Interval groups related to finite Coxeter groups I Open
We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type . Type is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations…
View article: Using edt0l systems to solve some equations in the solvable Baumslag-Solitar groups
Using edt0l systems to solve some equations in the solvable Baumslag-Solitar groups Open
We investigate the solution sets to equations in the solvable Baumslag-
\nSolitar groups BS(1, k), k ≥ 2, and show that these sets are represented
\nby edt0l languages in some cases. In particular, we prove that
\nthe multiplication table …
View article: Isomorphism and non-isomorphism for interval groups of type D
Isomorphism and non-isomorphism for interval groups of type D Open
We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group W(Dn). We use combinatorial methods to derive alternative presentations for the groups, and use t…
View article: The Artin monoid Cayley graph
The Artin monoid Cayley graph Open
In this paper we investigate properties of the Artin monoid Cayley graph. This is the Cayley graph of an Artin group $A_Γ$ with respect to the (infinite) generating set given by the associated Artin monoid $A^+_Γ$. In a previous paper, the…
View article: Isomorphism and non-isomorphism for interval groups of type D_n
Isomorphism and non-isomorphism for interval groups of type D_n Open
We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentation…
View article: Interval groups related to finite Coxeter groups, Part II
Interval groups related to finite Coxeter groups, Part II Open
We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associ…
View article: The development of the theory of automatic groups
The development of the theory of automatic groups Open
We describe the development of the theory of automatic groups. We begin with a historical introduction, define the concepts of automatic, biautomatic and combable groups, derive basic properties, then explain how hyperbolic groups and the …
View article: Using EDT0L systems to solve some equations in the solvable Baumslag-Solitar groups
Using EDT0L systems to solve some equations in the solvable Baumslag-Solitar groups Open
We investigate the solution sets to equations in the solvable Baumslag-Solitar groups $BS(1,k)$, $k\geq2$, and show that these sets are represented by EDT0L languages in some cases. In particular, we prove that the multiplication table of …
View article: Automaticity for graphs of groups
Automaticity for graphs of groups Open
In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associat…
View article: Interval groups related to finite Coxeter groups I
Interval groups related to finite Coxeter groups I Open
We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The present…
View article: The compressed word problem in relatively hyperbolic groups
The compressed word problem in relatively hyperbolic groups Open
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
View article: Biautomatic structures in systolic Artin groups
Biautomatic structures in systolic Artin groups Open
We examine the construction of Huang and Osajda that was used in their proof of the biautomaticity of Artin groups of almost large type. We describe a slightly simpler variant of that biautomatic structure, with explicit descriptions of a …
View article: Multifraction reduction IV: Padding and Artin-Tits groups of sufficiently large type
Multifraction reduction IV: Padding and Artin-Tits groups of sufficiently large type Open
We investigate the padded version of reduction, an extension of multifraction reduction as defined in arXiv:1606.08991, and connect it both with ordinary reduction and with the so-called Property $\mathrm{H}$. As an application, we show th…
View article: A method for building permutation representations of finitely presented groups
A method for building permutation representations of finitely presented groups Open
We design an algorithm to find certain partial permutation representations of a finitely presented group $G$ (the bricks) that may be combined to a transitive permutation representation of $G$ (the mosaic) on the disjoint union.
View article: A method for building permutation representations of finitely presented\n groups
A method for building permutation representations of finitely presented\n groups Open
We design an algorithm to find certain partial permutation representations of\na finitely presented group $G$ (the bricks) that may be combined to a\ntransitive permutation representation of $G$ (the mosaic) on the disjoint\nunion.\n
View article: Rapid decay and Baum-Connes for large type Artin groups
Rapid decay and Baum-Connes for large type Artin groups Open
We prove that many Artin groups of large type satisfy the rapid decay property, including all those of extra-large type. For many of these, including all 3-generator groups of extra-large type, a result of Lafforgue applies to show that th…
View article: The generalised word problem in hyperbolic and relatively hyperbolic\n groups
The generalised word problem in hyperbolic and relatively hyperbolic\n groups Open
We prove that, for a finitely generated group hyperbolic relative to\nvirtually abelian subgroups, the generalised word problem for a parabolic\nsubgroup is the language of a real-time Turing machine. Then, for a hyperbolic\ngroup, we show…
View article: The generalised word problem for subgroups of hyperbolic groups
The generalised word problem for subgroups of hyperbolic groups Open
We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infin…