## GeoMeG: Description

The general goal of the project is to study Lie groups equipped with arbitrary distance functions and to develop an adapted geometric measure theory in the non-Riemannian settings.

The main focus is subRiemannian geometry, together with implications to control systems and nilpotent groups. SubRiemannian spaces, and especially Carnot groups, appear in various areas of mathematics, such as control theory, harmonic and complex analysis, subelliptic PDE's and geometric group theory.

The members of the team master methods coming from geometric measure theory, geometric analysis, and group theory; in particular, analysis on metric spaces and Lie group theory.

| Grant Agreement | University of Jyvskäylä Press Release |

## People

### Principal Investigator

Enrico Le Donne

| Personal website |

### Post-Docs

Sebastiano Don
Laurent Dufloux
Thibaut Dumont
Rami Luisto
Matthew Romney
Francesca Tripaldi

Fares Essebei
Eero Hakavuori
Ville Kivioja
Terhi Moisala

## Jyväskylä Geometry Seminar

The Jyväskylä Geometry seminar is on Mondays afternoons
in the Department of Mathematics and Statistics.
The meeting is usually from 14:15 to 16:00 in room MaD 380.

### Winter 2019

 21/January/2019 On BV Functions and Integration by Parts Formulæ in Metric Measure Spaces (MaD 380) Vito Buffa (Aalto University)   We give a characterization of BV functions in metric measure spaces making use of suitable vector fields. This constitutes the starting point for a discussion on Gauss-Green Formulæ featuring the normal traces of divergence-measure vector fields, and for new results on the traces of BV functions. Based on joint works with G. E. Comi and M. Miranda Jr.
 3/June/2019 TBA (MaD 380) David Fisher (Indiana University Bloomington)
 10/June/2019 TBA (MaD 380) David Freeman (University of Cincinnati Blue Ash College)

See all current Seminars | Analysis Seminar |

Past Seminars: 2018 | 2017 | (older) |

## Future Events

23rd Sep - 3rd Oct 2019 | Oxford (UK)
Pansu's Fest, for the 60th birthday of Pierre Pansu
| Website |

11th-16th August 2019 | Jyväskylä (Finland)
Jyväskylä Summer School - Courses in Geometry
| Website |

18th-22nd February 2019 | Jyväskylä (Finland)
Sub-Riemannian Geometry and Beyond, II
| Website |

 Past Events February/2018 Sub-Riemannian Geometry and Beyond Jyväskylä (Finland)

## Recent Papers

We warmly thank the CVGMT Team for the support and
for hosting the following papers at CVGMT server.

 Metric Lie groups admitting dilations (2019) E. Le Donne, S. Nicolussi Golo    We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,\infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors.
 Blowups and blowdowns of geodesics in Carnot groups (2018) E. Hakavuori, E. Le Donne    We study infinitesimal and asymptotic properties of geodesics (i.e., isometric images of intervals) in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. With the same approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result can be extended to rough geodesics.
 Restricting open surjections (2018) J. Jaramillo, E. Le Donne, T. Rajala   (Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas)    We show that any continuous open surjection from a complete metric space to another metric space can be restricted to a surjection for which the domain has the same density character as the target. This improves a recent result of Aron, Jaramillo and Le Donne.
 A note on topological dimension, Hausdorff measure, and rectifiability (2018) G. David, E. Le Donne    The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $H^n(X)$, is finite. Suppose further that the lower $n$-density of the measure $H^n(X)$ is positive, $H^n(X)$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $H^n(X)$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Csörnyei-Jones.
 Toward a quasi-Möbius characterization of Invertible Homogeneous Metric Spaces (2018) D. Freeman, E. Le Donne    We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Möbius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.
 On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups (2018) K. Fässler, E. Le Donne    This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric' is not a transitive relation.

| See the complete list of papers |

## Open Positions

Please, contact the PI if you are interested to do a Ph.D. or a postdoc
in the framework of GeoMeG ERC project.

New positions will be open in the next years.

Prof. Enrico Le Donne

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