We consider a combinatorial approximation of the hyperbolic plane, and show that the perimeter of a disk with radius in this model is given by . We use properties of Fibonacci numbers to show that this model satisfies a linear isoperimetric inequality.

For random groups in the Gromov density model at , we construct walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities  and , respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density .

A graph is hamiltonian-connected if every pair of vertices can be connected by a hamiltonian path, and it is hamiltonian if it contains a hamiltonian cycle. Every hamiltonian-connected graph is hamiltonian, however we also construct families of nonhamiltonian graphs with `many’ hamiltonian paths, where ‘many’ is interpreted with respect to the number of pairs of vertices connected by hamiltonian paths. We then consider minimal graphs that are hamiltonian-connected; we show that any order- graph that is hamiltonian-connected graphs must have  edges, and we construct an infinite family of graphs realizing this minimum.

The -gonal models of random groups are defined as the quotients of free groups on n generators by cyclically reduced words of length k. As  tends to infinity, this model approaches the Gromov density model. In this paper we show that for any fixed , if positive -gonal random groups satisfy Property (T) with overwhelming probability for densities , then so do -gonal random groups, for any . In particular, this shows that for densities above , groups in -gonal models satisfy Property (T) with probability 1 as  approaches infinity.

Introduced recently, an -crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single -crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type of übercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be concentrated in a permutation corresponding to the levels at which the strands lie within the crossing. These ideas give rise to two new invariants for a knot : the übercrossing number , and petal number . These are the least number of loops in any übercrossing or petal projection of , respectively. We relate and to other knot invariants, and compute for several classes of knots, including all knots of nine or fewer crossings.

An increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that and (where is a positive integer greater than ), among others, are universal. In a forthcoming paper, Adams introduces the notion of a multi-crossing projection of a knot. An -crossing projection is a projection of a knot in which each crossing has strands, rather than strands as in a regular projection. We then extend the notion of universality to such knots. These results allow us to prove that is a universal sequence for both n-crossing knot projections, for all . Adams further proves that all knots have an -crossing projection for all positive . Another proof of this fact is included in this paper. This is achieved by constructing -crossing template knots, which enable us to construct multi-crossing projections with crossings of any multiplicity.