We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

This is an expository article on discrete geometric analysis based on the lectures which the author gave at Gregynog Hall, University of Wales, as an activity of the Project “Analysis on graphs and its applications” in the Issac Newton Institute. Topics are selected to illustrate the nature of the field; that is, we focus upon analysis on graphs with a geometric flavor such as graph versions of harmonic theory and spectral geometry of Laplacians. Zeta functions and random walks are also discussed from a geometric viewpoint.

Discrete Geometric Analysis

In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S1-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.

Approximating L 2 -Invariants and Homology Growth

https://math.ucr.edu/~lapidus/papers/YJFAN_5356.pdf

Ihara's zeta function for periodic graphs and its approximation in the amenable case☆

We establish a new formula for the heat kernel on regular trees in terms of classical $$I$$
 -Bessel functions. Although the formula is explicit, and a proof is given through direct computation, we also provide a conceptual viewpoint using the horocyclic transform on regular trees. From periodization, we then obtain a heat kernel expression on any regular graph. From spectral theory, one has another expression for the heat kernel as an integral transform of the spectral measure. By equating these two formulas and taking a certain integral transform, we obtain as application several generalized versions of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. Our approach to the Ihara zeta function and determinant formula through heat kernel analysis follows a similar methodology which exists for quotients of rank one symmetric spaces.

/pdf/heat-kernels-on-regular-graphs-and-generalized-ihara-zeta-51pegioaac.pdf

Heat kernels on regular graphs and generalized Ihara zeta function formulas

Zeta functions of discrete groups acting on trees

Every fall, millions of Americans enter betting pools to pick winners of the weekly NFL football games. In the spring, NCAA tournament basketball pools are even more popular. In both cases, teams that are popularly perceived as “favorites” gain a disproportionate share of entries. In large pools there can be a significant advantage to picking upsets that differentiate your picks from the crowd.

In this paper, we present a model of betting pools that incorporates pool participant behavior. We use the model to derive strategies that maximize the expected return on a bet in both football pools and tournament-style pools. These strategies significantly outperform strategies based on maximizing score or number of correct picks---often by orders of magnitude.

https://cs.slu.edu/~clair/preprints/pools-or.pdf

Optimal Strategies for Sports Betting Pools

The L 2 -zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y, the normalized zeta functions of the finite graphs converge to the L 2 -zeta function of Y.

/pdf/convergence-of-zeta-functions-of-graphs-3o2tcp8an5.pdf

Convergence of zeta functions of graphs

Growth of Betti numbers

This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate Hilbert representation is defined. Zeta functions are defined for infinite graphs with a cocompact or finite covolume group action.

Zeta Functions Of Discrete Groups Acting On Trees

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