It is well known that any torus bundle over the circle with Anosov monodromy and any torus semi-bundle (Sapphire space), which has a double cover homeomorphic to such a torus bundle, admit Sol structure. In this talk, I give an explicit construction of Sol structure of these manifolds, and then give a progress report on my project towards determination of their isometry groups and classification of their involutions. Sakuma [1985] gave a classification of involutions on torus bundles, and Barreto-Goncalves-Vendruscolo [2016] gave a classification of free involutions on torus semi-bundles. Thus this project will give a complete classification of involutions on 3-manifolds admitting Sol structure.

Greene in 2011 conjectured that if a pair of links have homeomorphic double branched coverings, then either both are alternating or both are non-alternating. In this talk, I introduce an idea due to Boileau to resolve this conjecture for \(\pi\)-hyperbolic knots and then give a progress report on my study inspired by the idea.

Edmonds proved that every periodic knot has an invariant minimal Seifert surface. Tollefson proved that every strongly invertible knot has a pair of mutually disjoint minimal-genus Seifert surfaces whose union is preserved by the involution. In this talk, we will explain the following results: (a) An algorithm to construct an invariant Seifert surface for a given strongly invertible knot, and (b) the existence of a family of strongly invertible knots such that the gaps between the “invariant genera” and the genera are arbitrarily large.

The right-angled Artin group (RAAG) on a finite graph \(\Gamma\) with the vertex set \(V(\Gamma)\) and the edge set \(E(\Gamma)\) is the group given by the following presentation, where we employ the convention opposite to the usual one:

\(G(Γ) = \langle v \in V(Γ) \mid [v_i , v_j] = 1 \; \text{if} \; \{ v_i, v_j \} \not\in E(Γ) \rangle.\)

It is well-known that the edgeless graph \(\overline{K}_n\) with \(n\) vertices is a global obstruction in the following sense: for any finite graph \(\Gamma\), \(G(\overline{K}_n) \cong \mathbb{Z}^n\) can be embedded into \(G(\Gamma)\) if and only if \(\overline{K}_n\) can be realized as a full subgraph of \(\Gamma\). In this talk, we introduce a generalization of this obstruction theorem to linear forests. If time permits, we apply the generalization to the problem of embedding RAAGs into mapping class groups.

There are 17 plane crystallographic groups, and many of them have more than one fundamental domains. I describe the group presentation and the growth function for each of the fundamental domains of the 17 plane crystallographic groups.

Finding lattices in \(\rm{PU}(n,1)\) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space. One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle. In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

Agol proved that every pseudo-Anosov mapping torus of a surface, punctured along the singular points of the stable and unstable foliations, admits a canonical “veering” ideal triangulation. In this talk, I will describe the veering triangulations of the mapping tori of some pseudo-Anosov maps arising from Penner's construction.

We give a new infinite family of non-fibered 2-bridge knots whose knot groups are bi-orderable.