This is a transcript of my notes from Marc Bourdon’s lectures. They may not always accurately reflect the content of the lectures, especially in places I did not put down everything that was said, and some comments or details might be mine.
Author Archives: jraimbau
Ingredients and consequences of quasi-isometric rigidity of lattices in certain solvable Lie groups (Tullia Dymarz)
This is a transcript of my notes from Tullia Dymarz’s lectures. They may not always accurately reflect the content of the lectures, especially in places I did not put down everything that was said, and some comments or details might be mine.
Quasi-isometric rigidity of nonuniform lattices (Misha Kapovich)
This is a transcript of my notes from Misha Kapovich’s lectures. They may not always accurately reflect the content of the lectures, especially in places I did not put down everything that was said, and some mistakes might have been introduced.
Cost of amenable groups
The aim of these notes is to provide an introduction to Gaboriau’s paper « Sur le coût des relations d’équivalence et des groupes » and to gather in one place various arguments, occuring in diverse sources, to give a characterisation of amenable groups through their actions on probability spaces.
Notes on the Abért–Nikolov theorem on rank gradient and cost (notes by Holger Kammeyer after his own lecture)
These are notes of a talk on a theorem of Abért–Nikolov: Let \(\Gamma\) be a finitely generated group and let \((\Gamma_n)\) be a chain of finite index subgroups. Assume that the action of \(\Gamma\) on the boundary \(\partial T\) of the coset tree of \((\Gamma_n)\) is essentially free. Then the rank gradient of \(\Gamma\) with respect to the chain \((\Gamma_n)\) equals the cost of the action of \(\Gamma\) on \(\partial T\).
Universal \( L^2 \)-torsion, \( L^2 \)-Euler characteristic, Thurston norm and polytopes (Wolfgang Lück)
The aim of the talk is to define an invariant (« universal \( L^2 \)-torsion ») from which many others (usual \( L^2 \)-torsion, \( L^2 \)-Alexander invariant and Euler characteristic,…) can be derived, as well as the relations between them.
Classifying spaces for families and their finiteness properties (Brita Nucinkis)
Let \( G \) be a group and \( \mathcal F \) a family of subgroups such that:
- For all \( H \in \mathcal F, g \in G \) we have also \( H^g := gHg^{-1} \in \mathcal F _);
- For all \( H, H’ \in \mathcal F \) we have \( H \cap H’ \in \mathcal F \).
For example \( \mathcal F \) can be :
- The trivial subgroup;
- Finite subgroups;
- Virtually cyclic subgroups;
- Free abelian, nilpotent, … subgroups
A model for the classifying space \e E_{\mathcal F}G \) is then a CW-complex \( X \) with a \( G \)-action such that:
- If \( H \in \mathcal F \) then the subset \( X^H \) of points fixed by \( H \) is a contractible subcomplex of \( X \);
- Otherwise \( X^H \) is empty.
For example, if \( \mathcal F = \{\{e\}\} \) then \( E_{\mathcal F}G \) is the classifying space \( EG \) for \( G \). The classifying space can also be defined by the following universal property: it is the only G-complex \( Y \) such that for every \( G \)-action whose point stabilisers are in \( F \), there exists a \( G \)-map \( E_{\mathcal F}G \to Y \) (which is unique up to \( G \)-homotopy).
Analytic aspects of locally compact groups (Corina Ciobotaru)
The objects of interest in this talk are the unitary representations of a locally compact group \( G \). These are homomorphisms \( \pi: G \to \mathrm U(\mathcal H) \) where \( \mathcal U(\mathcal H \) is the group of unitary operators on a Hilbert space \( \mathcal H \). It will be required that they be continuous in the following sense: for every \( v \in \mathcal H \) the map \( G \to \mathcal H, g \mapsto \pi(g)v \) is continuous. Basic examples are the following:
- The trivial representation;
- The left-regular representation \( \lambda: G \to \mathcal U(L^2(G, \mu_{Haar})) \) acting by \( \lambda(g)f(x) = f(g^{-1}x) \).
\( L^2 \)-Alexander torsions of 3–manifolds (Yi Liu)
Let \( N \) be an irreducible, compact, orientable 3–manifold whose boundary is either empty or contains only tori as connected components. Call a triple \( (G, \gamma, \phi) \) admissible if \( G \) is a discrete group, \( \gamma : \pi_1(N) \to G \) and \( phi : \pi = \pi_1(N) \to {\mathbb Z} \) are homomorphisms such that there exists a commutative diagram:
\[
\begin{array}{ccc}
\pi_1(N) & \overset{\gamma}{\rightarrow} & G \\
& \underset{\phi}{\searrow} & \downarrow \\
& & \mathbb Z \end{array}
\]
Fix a cellulation of \( N \) and let \( C_*(\widetilde N) \) be the chain complex of the universal cover. Let \( t > 0 \) and define a representation:
\[
\kappa(\gamma, \phi, t) :\left\{ \begin{array}{ll}
{\mathbb Z} \pi \to {\mathbb R} G \\
g \mapsto t^{\phi(g)}\gamma(g)
\end{array} \right.
\]
with which the twisted \( L^2 \)-complex \( \ell^2(G) \otimes_{\kappa(\gamma, \phi, t)} C_*(\widetilde N) \). Let \( \tau^{(2)}(N; \gamma, \phi)(t) \) be the \( L^2 \)-torsion of this complex in the case where it is well-defined (when the complex is weakly acyclic and all its differentials of determinant class) and 0 otherwise. Consider this construction as associating to the admissible triple \( (G, \gamma, \phi) \) a function \( tau^{(2)}(N; \gamma, \phi): ]0, +\infty [ \to [0, +\infty[ \).
Rank, combinatorial cost and homology growth of higher-rank lattices (Miklós Abért)
Rank gradient
For any finitely generated group \( H \) let \( d(H) \) be its rank, the minimal number of elements needed to generate \( H \). If \( H \) is a finite-index subgroup in a finitely generated group \( \Gamma \) then we have
\[
d(H) \le |\Gamma / H| (d(\Gamma) – 1)
\]
and it is thus natural to define:
\[
r(\Gamma, H) = \frac{d(H) – 1}{|\Gamma / H|}.
\]
If \( \Gamma = \Gamma_0 \supset \Gamma_1 \supset \cdots \) is a chain of finite index subgroup then the limit:
\[
\mathrm{RG}(\Gamma, \Gamma_n) = \lim_{n\to+\infty} r(\Gamma, \Gamma_n)
\]
exists, and is called the rank gradient of \( (\Gamma, (\Gamma_n)) \).
If \( (\Gamma_n), (\Delta_n) \) are two residual chains in the same group \( \Gamma \) (chains with \( \Gamma_n, \Delta_n \) normal in \( \Gamma \) and \( \bigcup_n \Gamma_n = \{ 1 \} = \bigcup_n \Delta_n \)), then are \( \mathrm{RG}(\Gamma, (\Gamma_n) \) and \( \mathrm{RG}(\Gamma, (\Delta_n)) \) equal?