Related papers: On Andrews-Curtis conjectures for soluble groups

The Andrews-Curtis conjecture asserts that, for a free group $F_n$ of rank $n$ and a free basis $(x_1,...,x_n)$, any normally generating tuple $(y_1,...,y_n)$ is Andrews-Curtis equivalent to $(x_1,...,x_n)$. This equivalence corresponds to…

The Andrews-Curtis conjecture remains one of the outstanding open problems in combinatorial group theory. It claims that every normally generating $r$-tuple of a free group $F_r$ of rank $r\geq 2$ can be reduced to a basis by means of…

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary…

The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced…

The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be transformed into the trivial presentation by a finite sequence of "elementary transformations" which are Nielsen transformations together with…

The paper discusses the Andrews-Curtis graph of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one them can be obtained from…

For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup…

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above…

We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of…

The generalized Andrews-Curtis Conjecture expects that finite PLCW 2-complexes which are simple-homotopy equivalent, can be 3-deformed into each other. If in addition subcomplexes are required to be kept fix during the deformation, this is…

We show that the Andrews-Curtis conjecture holds for all balanced presentations of the trivial group corresponding to Heegaard diagrams of $S^3$.

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

A conjecture of Roseberger asserts that every generalised triangle group either is virtually soluble or contains a non-abelian free subgroup. Modulo two exceptional cases, we verify this conjecture for generalised triangle groups of type…

It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in…

We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting…

Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new…

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of…

A conjecture of Dehornoy claims that, given a presentation of an Artin-Tits group, every word that represents the identity can be transformed into the trivial word using the braid relations, together with certain rules (between pairs of…

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties.…