This colloquium-style seminar will be an opportunity for me to present my field Geometric Group Theory (GGT) and some of my personal interests therein. The influence of Klein and his Erlangen's program lead geometers to look at geometries and their transformations from a group theoretic viewpoint. The study of a metric space and its group of isometries are closely related. A metric property on one side can dictate an algebraic one on the other, and vice-versa. The central idea of Geometric Group Theory is to look at groups themselves as geometric objects by endowing them with a distance function. The notion of nonpositive curvature in a metric space provide a rich framework for isometric group actions. Some classical examples will be discussed like symmetric spaces of Lie groups and their non-Archimedean counterparts: Euclidean buildings.

In the Euclidean space, by the celebrated result due to Bombieri, De Giorgi, and Miranda, all positive entire solutions of the minimal graph equation are constant. It turns out that on Riemannian manifolds similar results can be obtained for solutions with at most linear growth if the manifold has only one end and asymptotically non-negative sectional curvature. In this talk I will discuss about recent results concerning the existence and non-existence of entire minimal graphic and p-harmonic functions. Talk is based on joint work with Jean-Baptiste Casteras and Ilkka Holopainen.

Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. In this seminar, I present a recent work in collaboration with Ludovic Rifford where we show that the conjecture holds whenever the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and resolution of singularities.

The large scale geometry of Gromov hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the linear isoperimetric filling inequality for 1-cycles, the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. After briefly reviewing these properties, I will describe a number of closely analogous results for spaces of rank n > 1 in an asymptotic sense, under some weak assumptions reminiscent of non-positive curvature. A central role is played by a suitable notion of n-dimensional quasi-minimizing surfaces of polynomial growth of order n, which serve as a substitute for quasi-geodesics.

Box spaces are geometric objects associated to residually finite groups . They can be traced back to the 70's when Margulis proved (in modern formulation) that the box space of a residually finite group with Kazdhan's property (T) is a family of expander graphs. Since then, a lot of group properties have found counterparts in metric properties of their box-spaces. Property A, introduced by Guoliang Yu in connection with the coarse Baum-Connes conjecture, is a form of geometric amenability. We study whether that property for a residually finite group can be detected at the level of the box-space. We propose two different weakening of Property A, Property A at infinity and Fibred Property A to serve that purpose. Conjecturally, these properties should reflect $C^*$-exactness of the boundary groupoid of the space.

Large scale geometry deals with asymptotic properties of metric spaces, more formally with quasi-isometry invariants. Therefore, by the lemma of Svarc-Milnor, it plays an important role in geometric group theory. But it is interesting on its own, for example in the context of Riemannian geometry. In my talk I will focus on nilpotent Lie groups equipped with left-invariant Riemannian metrics and large scale analogues of isoperimetric inequalities.

Part 1: Asymptotic nilpotency of classical cellular automata A cellular automaton (CA) is a continuous shift-commuting self-map of a subshift. By a classical CA we refer to a CA defined on a full shift over a free abelian group, namely on $S^{\mathbb{Z}^d}$ for a finite alphabet $S$ under the shift action of $\mathbb{Z}^d$. For $d = 1$, it was shown in [Guillon-Richard, 2008] that if such a CA is asymptotically nilpotent (every configuration converges to the all-zero configuration), then actually every point maps to the all-zero point after a bounded number of steps. This was generalized to CA on free abelian groups by the author in 2012. We explain some of the geometric ideas behind the proofs, and discuss the importance of nilpotency in the decidability theory of CA. $$ \, $$ Part 2: Asymptotic nilpotency of non-classical cellular automata We call a (topological dynamical) $\mathbb{N}$-system asymptotically nilpotent (AN) if there is a special point 0 such that every point converges to 0 in the iteration of the dynamics, and uniformly asymptotically nilpotent (UAN) if this convergence is uniform over the space. A non-uniformly asymptotically nilpotent (NUAN) $\mathbb{N}$-system is one that is AN but not UAN. We say a family of $\mathbb{N}$-systems is nilrigid if it does not contain a NUAN map, and if $G$ is a group, we say a $G$-system is nilrigid if its family of endomorphisms ($G$-commuting continuous self-maps) is nilrigid. The results above state essentially that full shifts over abelian groups are nilrigid. We discuss how robust/fragile this result is, by studying nilrigidity when the "full shift" assumption is replaced by dynamical assumptions, e.g. by SFTness or soficness, when the alphabet is replaced by a manifold, or when we step into the measurable category, finding plenty of non-nilrigid situations. We then discuss the case when the acting group is non-abelian, e.g. the Heisenberg group or a free group.

Motivated by the work of Colding and Minicozzi and Hoffman and White on minimal laminations obtained as limits of sequences of properly embedded minimal disks, Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. Bernstein and Tinaglia prove that an embedded minimal surface $\Sigma\subset\Omega$ with the simple lift property must have genus zero, if $\Omega$ is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that $\Omega$ cannot contain closed minimal surfaces. In this work, I generalise this result by taking an arbitrary orientable three-manifold $\Omega$ and proving that one is able to restrict the topology of an arbitrary surface $\Sigma\subset\Omega$ with the simple lift property. Among other things, I prove that the only possible compact surfaces with the simple lift property are the sphere and the torus in the orientable case, and the connected sum of up to four projective planes in the non-orientable case. In the particular case where $\Sigma\subset\Omega$ is a leaf of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks, we are able to sharpen the previous result, so that the only possible compact leaves are the torus and the Klein bottle.