Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let $$\Sigma _{g}$$ Σ g denote a topological surface of genus $$g\ge 2$$ g ≥ 2 . We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) under a random representation of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) into $$\mathsf {SU}(n)$$ SU ( n ) . Each such expected value involves a contribution from all irreducible representations of $$\mathsf {SU}(n)$$ SU ( n ) . The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

A fibering theorem for 3-manifolds

We generalize a result of Moon on the fibering of certain 3-manifolds over the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold. Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infinite index in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$ of $G$, and suppose that $N$ has a composition series of length $n$ in which at least $n-1$ terms are finitely generated. Suppose that $N$ intersects nontrivially the fundamental groups of the splitting tori given by the Geometrization Theorem and that the intersections of $N$ with the fundamental groups of the geometric pieces are non-trivial and not isomorphic to $\mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $\mathbb{S}$ with fiber a compact surface $F$ such that $\pi_1(F)$ and $U$ are commensurable.

Folding-like techniques for CAT(0) cube complexes

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

Free quotients of fundamental groups of smooth quasi-projective varieties

We study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

Deformations of circle-valued Morse functions on 2-torus

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued Morse functions on T2 with respect to the action of the group of diffeomorphisms of T2

Computing the fundamental group of a higher-rank graph

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.