# GeoMeG Geometry of Metric groups (GeoMeG)

Main funder

Funder's project number: 713998

Funds granted by main funder (€)

- 1 248 560,00

Funding program

- ERC Starting Grant (ERC European Research Council,...)

Project timetable

Project start date: 01/08/2017

Project end date: 20/10/2020

Summary

What are the best trajectories to park a truck with several trailers? How fast can a lattice grow? These questions are some of the ones studied in this project because both the infinitesimal structure of controlling a truck and the asymptotic geometry of a (nilpotent) lattice are examples of metric groups: Lie groups with homogeneous distance. The PI plans to study geometric properties of metric groups and implications to control systems and nilpotent groups. In particular, the plan is to exploit the relation between the regularity of distinguished curves, sets, and maps in subRiemannian groups, volume asymptotics in nilpotent groups, and embedding results. SubRiemannian spaces, and in particular Carnot groups, appear in various areas of Mathematics, such as Control Theory, Harmonic and Complex Analysis, subelliptic PDE´s and Geometric Group Theory. The results in this project will provide more links between such areas. The general goal is to develop an adapted Geometric Measure Theory. The PI developed a net of high level international collaborations and obtained several results via a combination of analysis on metric spaces (differentiation of Lipschitz maps, tangents of measures, and Gromov-Hausdorff limits) and the theory of locally compact groups (Lie group techniques and the solutions of the Hilbert 5th problem). This allowed th PI to solve a number of open problems in the filed, as the analogue of Myers-Steenrod Theorem on the smoothness of isometries,

of Nash Isometric Embedding and recently the non-minimality of curves with corners. Some of the next aims are to establish an analogue of the De Giorgi´s rectifiability result for finite-perimeter sets and prove the smoothness of geodesics, a 30-year-old open problems. Many more related questions are still open and the goal of this project is to tackle them.

of Nash Isometric Embedding and recently the non-minimality of curves with corners. Some of the next aims are to establish an analogue of the De Giorgi´s rectifiability result for finite-perimeter sets and prove the smoothness of geodesics, a 30-year-old open problems. Many more related questions are still open and the goal of this project is to tackle them.

Principal Investigator

Primary responsible unit

Related publications

- On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups (2021) Fässler, Katrin; et al.; A1; OA
- A note on topological dimension, Hausdorff measure, and rectifiability (2020) David, Guy C.; et al.; A1; OA
- Extending an example by Colding and Minicozzi (2020) Ruffoni, Lorenzo; et al.; A1; OA
- On the topology of surfaces with the generalised simple lift property (2020) Tripaldi, Francesca; A1; OA
- Open and discrete maps with piecewise linear branch set images are piecewise linear maps (2020) Luisto, Rami; et al.; A1; OA
- Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces (2020) Antonelli, Gioacchino; et al.; A1; OA
- Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems (2020) Sachkov, Yuri L.; A1; OA
- Stoïlow's theorem revisited (2020) Luisto, Rami; et al.; A1; OA
- A Newman property for BLD-mappings (2019) Luisto, Rami; A1; OA
- A Sub-Finsler Problem on the Cartan Group (2019) Ardentov, A. A.; et al.; A1; OA