Related papers: The balanced superelliptic mapping class groups ar…
We investigate an extended version of the stable Andrews-Curtis transformations, referred to as EAC transformations, and compare it with a notion of equivalence in a family of $3$-manifolds with boundary, called the {\emph{simple balanced…
Let $\textrm{Mod}(N_{g, p})$ denote the mapping class group of a nonorientable surface of genus $g$ with $p$ punctures. For $g\geq14$, we show that $\textrm{Mod}(N_{g, p})$ can be generated by five elements or by six involutions.
A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a…
We describe the outer automorphism group of a one-ended fundamental group of a graph of groups, when edge groups are cyclic, and vertex groups are torsion-free with cyclic centralizers. We show that in this case the outer automorphism group…
We say that a subset $X$ quasi-isometrically boundedly generates a finitely generated group $\Gamma$ if each element $\gamma$ of a finite-index subgroup of $\Gamma$ can be written as a product $\gamma = x_1 x_2 \cdots x_r$ of a bounded…
We describe the action of the mapping class group $M(g,n)$ on the fundamental group of $T_{g,n}$, a compact orientable topological surface of positive genus $g$ with $n$ marked points. This is achieved by computing the image of the…
We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of…
Mann and Rafi's seminal work initiated the study of the coarse geometry of big mapping class groups. Specifically, they construct coarsely bounded (CB) generating sets for mapping class groups of a large class of infinite-type surfaces. In…
We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at…
A gain graph is a triple (G,h,H), where G is a connected graph with an arbitrary, but fixed, orientation of edges, H is a group, and h is a homomorphism from the free group on the edges of G to H. A gain graph is called balanced if the…
We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice…
The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman…
The Torelli group, I(S_g), is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying I(S_g): bounding pair maps,…
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for…
Monodromy groups, i.e. the groups of isometries of the intersection lattice L_X:=H_2/torsion generated by the monodromy action of all deformation families of a given surface, have been computed in math.AG/0006231 for any minimal elliptic…
The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (non necessarily regular) Rees matrix semigroup over a group. The formula is expressed in…
We show that the semigroup associated to a second-order elliptic system is positive if and only if the differential equations are essentially decoupled and the coefficients are real-valued. This means the system can be replaced by an…
Motivated by a problem in graph theory, this article introduces an algebra called the balanced algebra. This algebra is defined by generators and relations, and the main goal is to find a minimal set of relations for it.
Let $N_{g,n}$ denote the closed non-orientable surface of genus $g$ with $n$ punctures and let ${\mathcal N}_{g,n}$ denote the mapping class group of $N_{g,n}$. Szepietowski showed that ${\mathcal N}_{g,n}$ is generated by finitely many…
Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple…