Lorenzo Ruffoni - The fundamental groups of 3-pseudomanifolds

Abstract

Talk

Abstract

The study of 3-manifold has been a major source of inspiration for geometric group theory, providing the initial motivation behind many definitions and theorems. More generally, 3-Pseudomanifolds are polyhedral complexes that look like 3-manifolds almost everywhere, but can have some singular vertices whose links are surfaces of positive genus. These spaces naturally arise in the study of certain Coxeter groups, but in general the class of fundamental groups of 3-pseudomanifolds remains quite mysterious. In this talk we will discuss similarities and differences between hyperbolic 3-manifolds and hyperbolic 3-pseudomanifolds, focusing on problems such as computing the L2-homology and finding nice isometric actions.

Talk

Goals

1. Invitation to $3$-pseudomanifolds
2. $L^2$-homology from a geometric pov

1. Definitions

Definition

A $3$-Pseudomanifold $P$ is a $3$-dim polyhedral complex s.t. at every vertex the links is a surface. (If all links are $S^2$, then we get a normal $3$-mfd.)

Think of it as: $P$ is a $3$ manifold outside of some vertices.

Example

From now on $N$ will be a compact $3$-mfd with boundary. We cone off $N$, meaning, we add a cone to the $2$-dim boundary. We can go back by cutting puncturing the end of the cone.
This capping off gives us a pseudo-manifold $P$.

Main example

Let $W_L$ be a Coxeter group defined as follows: We have one generator for every vertex of some graph of order 2. Two elements commute if they are connected by an edge. There is something called a David complex, which is a CAT(0)-cube complex, which is acted upon by the Coxeter group.

We can show, that RACG are virtually fundamental groups of pseudo-mfds. Coxeter groups are interesting, because they algebraically catch the combinatorics of a graph.

Lemma

An easy lemma allows us to calculate the Euler characteristic by summing over the links

Consider $M,P$ of negative curvature (locally CAT(-1) metric).

	$\chi$	actions on $\mathbb{H}$	actions on CAT(0) ccx	Boundary	L2-homology	Fibering	Coherent
$M$ 3-mfd	$0$	yes	yes	$S^2$	$=0$	Yes	Yes
$P$ 3-pmfd	$>0$	yes/no	yes/no	$P$ Pontryagin sphere	$\ne 0$	Conjecturally yes. Currently only algebraic fibering	conj. no
Radhika Gupta did some research on this topic. (Field-Gupta-Lynon-Stark)
We want to understand the yes/no parts.

Theorem (Dobra, Lee, Marquis '25)

There is a $P$ and an inclusion of the Fundamental group of $P$ into the fundamental group of a 120-cell (closed ${\mathbb{H}}^4$-mdf)

Theorem (Manning-R.)

There is a $P$ such that the fundamental group cannot act properly on ${\mathbb{H}}^n,{\mathbb{H}}_{\mathbb{C}}^n,CAT(0)$ ccx.

2. $L^2$-homology of $P$

Theorem (R-Walsh)

Let $P$ be a compact cube cx. s.t. for all singular vertices the link is a flag and locally consists of at least $7$-triangles that form a surface. Flag complex. Then the L2-Betti numbers vanish except for $k=2$ where it is equal to the Euler characteristic.

For 3-manifolds the $l^2$-Betti numbers vanish except in some dimensions. Kielak and Gaboriau did some work on this topic. They showed that the fundamental group of $P$ is virtually Algebraically fibered group and not a Coherent group.

Similar work has been done by: Jankiewicz, Norin, Wise, Italiano, Martelli, Miglionni, Lafont, Minemyr, Sorcar, Stover, Wells, Isakovic. (Looks like a very active branch of research).

We just presented a picture of how to compute $l^2$ for a tree. This looks pretty nice. Sadly, I couldn’t follow.

There is some connection with Orbifold