Related papers: The generalised word problem for subgroups of hype…
The Boone--Higman conjecture is that every recursively presented group with solvable word problem embeds in a finitely presented simple group. We discuss a brief history of this conjecture and work towards it. Along the way we describe some…
In 2018, it was shown that all finitely generated virtually Abelian groups have multiple context-free word problems, and it is still an open problem as to where to precisely place the word problems of hyperbolic groups in the formal…
We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear,…
We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes…
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their…
Given a finitely generated relatively hyperbolic group $G$, we construct a finite generating set $X$ of $G$ such that $(G,X)$ has the `falsification by fellow traveler property' provided that the parabolic subgroups $\{H_\omega\}_{\omega\in…
We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite…
Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite…
We prove that a hyperbolic group cannot contain a strictly ascending chain of free quasiconvex subgroups of constant rank.
We prove that an infinite-ended group whose one-ended factors have finite-index subgroups and are in a family of groups with a nonzero multiplicative invariant is not quasi-isometrically rigid. Combining this result with work of the first…
This work is concerned with the stable norm in word hyperbolic groups as defined by Gromov. We give a short elementary proof of one of its basic property, that is existence of a computable uniform non null lower bound for stable norm in a…
The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent…
We show that every word hyperbolic, surface-by-(noncyclic) free group Gamma is as rigid as possible: the quasi-isometry group of Gamma equals the abstract commensurator group Comm(Gamma), which in turn contains Gamma as a finite index…
A finitary automaton group is a group generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at an identity state. We show that, for this presentation of finite groups, the…
We present a new algorithm deciding if the intersection of a quasiconvex subgroup of a negatively curved group with a conjugate is finite. We also give a short proof of decidability of the membership problem for quasiconvex subgroups of…
William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for…
We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.
We prove that the class of finitely presented inverse monoids whose Sch\"utzenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Sch\"utzenberger automata are context-free.…
We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of…
A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually…