Vertex-Facet Incidences of Unbounded Polyhedra

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure

Scl, sails and surgery

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programming. Furthermore, we show that the scl spectrum of nonabelian free groups contains elements congruent to every rational number modulo $\mathbb{Z}$, and contains well-ordered sequences of values with ordinal type $\omega^\omega$. Finally, we study families of elements $w(p)$ in free groups obtained by surgery on a fixed element $w$ in a free product of Abelian groups of higher rank, and show that \scl(w(p)) \to \scl(w) as $p \to \infty$.Comment: 23 pages, 4 figures; version 3 corrects minor typo

Hurwitz ball quotients

We consider the analogue of Hurwitz curves, smooth projective curves $C$ of genus $g \ge 2$ that realize equality in the Hurwitz bound $|\mathrm{Aut}(C)| \le 84 (g - 1)$, to smooth compact quotients $S$ of the unit ball in $\mathbb{C}^2$. When $S$ is arithmetic, we show that $|\mathrm{Aut}(S)| \le 288 e(S)$, where $e(S)$ is the (topological) Euler characteristic, and in the case of equality show that $S$ is a regular cover of a particular Deligne--Mostow orbifold. We conjecture that this inequality holds independent of arithmeticity, and note that work of Xiao makes progress on this conjecture and implies the best-known lower bound for the volume of a complex hyperbolic $2$-orbifold.Comment: Several improvements incorporating referee's comments. To appear in Math.

Clusters of Cycles

A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar 2--co nnected finite or countable graph $G$ of girth $r$ and maximal vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the plane, denote it by $P(G)$, i.e. such that: (i) all interior vertices are of degree $q$, (ii) all interior faces (denote their number by $p_r$) are combinatorial $r$-gons and (implied by (i), (ii)) (iii) all vertices, edges and interior faces form a cell-complex. An example of $(r,q)$-polycycle is the skeleton of $(r^q)$, i.e. of the $q$-valent partition of the sphere $S^2$, Euclidean plane $R^2$ or hyperbolic plane $H^2$ by regular $r$-gons. Call {\it spheric} pairs $(r,q)=(3,3),(3,4),(4,3),(3,5),(5,3)$; for those five pairs $P(r^q)$ is $(r^q)$ without the exterior face; otherwise $P(r^q)=(r^q)$. We give here a compact survey of results on $(r,q)$-polycycles.Comment: 21. to in appear in Journal of Geometry and Physic

Stable Commutator Length in Amalgamated Free Products

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed varies quasirationally in the orders of the free factors.Comment: 28 pages, 5 figures. Corrected typographical errors, changed exposition, added new results on quasirationalit