Related papers: Nielsen equivalence in closed surface groups

We prove that any generating tuple of the fundamental group of a sufficiently large 2-dimensional orbifold is represented by an almost orbifold covering. As a corollary we obtain a generalization of Louder's Theorem which asserts that any…

We show that there are infinitely many Nielsen equivalence classes of the mapping class group of a closed oriented surface of genus at least eight.

In this paper we give a complete classification of minimal generating systems in a very general class of Fuchsian groups G. This class includes for example any G which has at least seven non-conjugate cyclic subgroups of order greater than…

We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural…

We use the geometry of the Farey graph to give an alternative proof of the fact that if $A \in GL_2\mathbb Z$ and $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ is generated by two elements, there is a single Nielsen equivalence class of $2$-element…

Let $G \simeq M \rtimes C$ be an $n$-generator group with $M$ Abelian and $C$ cyclic. We study the Nielsen equivalence classes and T-systems of generating $n$-tuples of $G$. The subgroup $M$ can be turned into a finitely generated faithful…

For a group $G$ generated by $k$ elements, the Nielsen equivalence classes are defined as orbits of the action of $\operatorname{Aut} F_k$, the automorphism group of the free group of rank $k$, on the set of generating $k$-tuples of $G$.…

Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that…

We extend the result of \cite{Dutra} to generating pairs of triangle groups, that is, we show that any generating pair of a triangle group is represented by a special almost orbifold covering.

L. Louder showed that any generating tuple of a surface group is Nielsen equivalent to a stabilized standard generating tuple i.e. $(a_1,\ldots ,a_k,1\ldots, 1)$ where $(a_1,\ldots ,a_k)$ is the standard generating tuple. This implies in…

We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are…

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case.…

We determine the generators of the autoequivalence group of the derived category of coherent sheaves on a bielliptic surface over an algebraically closed field of arbitrary characteristic. As a consequence, we prove that any algebraic…

Motivated by the recent work of Algom-Kfir and Bestinva introducing the mapping class group of an infinite graph via proper homotopy equivalences, we give a necessary and sufficient condition for a surface to be properly homotopy equivalent…

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n,…

We define dual equivalence for any collection of combinatorial objects endowed with a descent set, and we show that giving a dual equivalence establishes the symmetry and Schur positivity of the quasi-symmetric generating function. We give…

Here we show that a finite nilpotent group is 2-closed if and only if it is either cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.

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…

We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.

For a closed surface $S$, its Torelli group $\mathcal{I}(S)$ is the subgroup of the mapping class group of $S$ consisting of elements acting trivially on $H_1(S;\mathbb{Z})$. When $S$ is orientable, a generating set for $\mathcal{I}(S)$ is…