### Session S23 - Group actions in Differential Geometry

## Talks

Wednesday, July 14, 11:00 ~ 11:30 UTC-3

## Manifold submetries, with applications to Invariant Theory

### Marco Radeschi

#### University of Notre Dame, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given an orthogonal representation of a Lie group G on a Euclidean vector space V, Invariant Theory studies the algebra of G-invariant polynomials on V. This setting can be generalized by replacing the orbits of the representation with a foliation by the fibers of a manifold submetry from the unit sphere S(V), and consider the algebra of polynomials that are constant along these fibers (effectively producing an Invariant Theory, but without groups). In this talk we will exhibit a surprisingly strong relation between the geometric information coming from the submetry and the algebraic information coming from the corresponding algebra, with several applications to classical Invariant Theory.

Joint work with Ricardo Mendes.

Wednesday, July 14, 11:40 ~ 12:10 UTC-3

## Lie theory for groupoids and compatible geometries

### Alejandro Cabrera

#### Universidade Federal do Rio de Janeiro, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we will first overview the Lie theory of groupoids and algebroids and then summarize some recent results focused on explicit constructions. We also detail applications to the study of geometric structures including Poisson brackets and their symplectic realizations. Most of these results were obtained in collaboration with I. Marcut and M. A. Salazar.

Wednesday, July 14, 12:20 ~ 12:50 UTC-3

## Prescribing Ricci curvature on a product of spheres

### Anusha Krishnan

#### Syracuse University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a symmetric 2-tensor T on a manifold M, does there exist a Riemannian metric g such that Ric(g) = T? I will discuss some classical results as well as some recent work in the presence of symmetry.

Joint work with TImothy Buttsworth (University of Queensland, Australia).

Wednesday, July 14, 14:00 ~ 14:30 UTC-3

## Infinitesimal maximal symmetry for some solvmanifolds

### Michael Jablonski

#### University of Oklahoma, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Recently, it was shown that Einstein metrics on solvmanifolds enjoy the special property that their isometry groups are the largest possible isometry groups on those homogeneous spaces. While the same result cannot hold for Ricci solitons, one can show that these special metrics do have the largest possible isometry algebra. We will discuss this result and the question of maximal symmetry for solvmanifolds more generally.

Joint work with Carolyn Gordon (Dartmouth College).

Wednesday, July 14, 14:40 ~ 15:10 UTC-3

## Submanifolds of Noncompact Homogeneous Spaces with Special Curvature Properties

### Megan Kerr

#### Wellesley College, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Ricci curvature form of a submanifold is not, in general, the restriction of the Ricci curvature of the ambient space. Therefore, classes of manifolds and submanifolds where the Ricci curvatures are aligned are very special. Indeed, Tamaru exploited this idea in the setting of noncompact symmetric spaces to construct new examples of Einstein solvmanifolds via special subalgebras. We characterize the largest category in which Tamaru's construction can be extended, identifying two crucial algebraic/metric conditions. We explore a new class of solvmanifolds defined by Kac-Moody algebras that are generalizations of symmetric spaces for which our crucial extra conditions hold. And furthermore, in current work in progress, we investigate other metric properties of these spaces.

Joint work with Tracy L. Payne (Idaho State University).

Wednesday, July 14, 15:20 ~ 15:50 UTC-3

## Minimal 2-spheres in ellipsoids of revolution

### Renato Bettiol

#### CUNY (Lehman College), USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Motivated by Morse-theoretic considerations, Yau asked in 1987 whether all minimal 2-spheres in a 3-dimensional ellipsoid inside $\mathbb R^4$ are planar, i.e., determined by the intersection with a hyperplane. Recently, this was shown not to be the case by Haslhofer and Ketover, who produced an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, combining Mean Curvature Flow and Min-Max methods. Using Bifurcation Theory and the symmetries that arise if at least two semi-axes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with P. Piccione.

Joint work with Paolo Piccione (Universidade de Sao Paulo, Brazil).

Friday, July 16, 16:00 ~ 16:30 UTC-3

## On some non-canonical abelian almost 3-contact structures on Lie groups

### Adrián Andrada

#### Universidad Nacional de Córdoba - CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In a recent work with G. Dileo [AD], we introduced the notion of \textit{abelian} almost 3-contact structure on Lie groups, and studied their algebraic and geometrical properties. Among these structures we highlighted a particular class, called canonical, which admit a compatible metric together with a metric connection with skew-symmetric torsion.

The aim of this talk is to introduce a family of non-canonical abelian almost 3-contact structures, which are closely related to 2-step nilpotent Lie groups with left invariant hypercomplex structures. The Lie groups carrying these structures are not solvable nor semisimple, and we show that some of them admit co-compact discrete subgroups. We study geometrical properties of some compatible metrics.

[AD] A. Andrada, G. Dileo, Odd dimensional counterparts of abelian complex and hypercomplex structures, to appear in \textit{Math. Nachr.}

Friday, July 16, 16:40 ~ 17:10 UTC-3

## Killing forms on nilpotent Lie groups

### Viviana del Barco

#### Universidade Estadual de Campinas, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

Killing forms on Riemannian manifolds are differential forms whose covariant derivative with respect to the Levi-Civita connection is skew-symmetric. They generalize to higher degree the concept of Killing vector fields. Examples of Riemannian manifolds with non-trivial Killing k-forms are quite rare for k≥2. Nevertheless they appear, for instance, on nearly-Kähler manifolds, round spheres and Sasakian manifolds. The aim of this talk is to introduce recent results regarding the structure of 2-step nilpotent Lie groups endowed with left-invariant Riemannian metrics and carrying non-trivial Killing forms. In the way, we will review aspects of the Riemannian geometry of nilpotent Lie groups endowed with left-invariant metrics and describe the methods to achieve the structure results.

Joint work with Andrei Moroianu, CNRS-Université Paris Saclay, France.

Friday, July 16, 17:20 ~ 17:50 UTC-3

## The geodesic flow on Lie groups

### Gabriela Paola Ovando

#### Universidad Nacional de Rosario and CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

The goal is the study of the integrability of the geodesic flow on Lie groups, when equipped with a left-invariant metric. We study Liouville integrability, specially on nilpotent Lie groups of steps two and three. It is shown that complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields. This holds for several families in dimensions up to six. Several algebraic relations on the Lie algebra of first integrals can be explicitly written. Also invariant first integrals are analyzed and several involution conditions are shown. With time, we shall see some kind of generalizations.

Friday, July 16, 19:00 ~ 19:30 UTC-3

## The classification of ERP $G_2$-structures on Lie groups

### Marina Nicolini

#### Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

On a differentiable 7-manifold, a $G_2$-structure $\varphi$ is a differentiable 3-form satisfying certain positivity condition. A closed $G_2$-structure (i.e. $d\varphi=0$) is called extremally Ricci pinched (ERP) if $d\tau=\frac{1}{6}|\tau|^2\varphi+\frac{1}{6}\ast(\tau\wedge\tau)$, where $\tau$ is the torsion 2-form of $\varphi$. There were only two known examples of ERP $G_2$-structures, one given by Bryant and the other one by Lauret, both homogeneous.

We first proved that some strong structural conditions must hold on the Lie algebra for the existence of ERP structures. Secondly, by using such a structural theorem we have obtained a complete classification of left-invariant ERP structures on Lie groups, up to equivalence and scaling. There are five of them, they are defined on five different completely solvable Lie groups and the $G_2$-structure is exact in all cases except one, given by the only example in which the Lie group is unimodular.

Joint work with Jorge Lauret (Universidad Nacional de Córdoba).

Friday, July 16, 19:40 ~ 20:10 UTC-3

## $SO(2)\times SO(3)$-invariant Ricci solitons and ancient flows on $\mathbb{S}^4$

### Timothy Buttsworth

#### The University of Queensland, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.

A full understanding of the behaviour of a Ricci flow near a singularity typically requires a classification of the possible Ricci solitons that can occur as singularity models. Such a classification exists for three-dimensional manifolds, but remains elusive for higher-dimensional manifolds. In this talk, I will describe recent attempts to understand the possible gradient shrinking Ricci solitons which can occur on $\mathbb{S}^4$ that are invariant under the usual group action of $SO(2)\times SO(3)$. It appears that the only such solitons are round, but our analysis also reveals the existence of a somewhat novel $\kappa$-noncollapsed ancient Ricci flow on $\mathbb{S}^4$.

Friday, July 16, 20:20 ~ 20:50 UTC-3

## Mountain pass approach to the prescribed Ricci curvature problem

### Artem Pulemotov

#### The University of Queensland, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The talk will focus on the prescribed Ricci curvature problem for homogeneous Riemannian metrics. We will discuss new results based on the variational interpretation of this problem and mountain pass techniques.

Joint work with Wolfgang Ziller (The University of Pennsylvania).

Friday, July 23, 16:00 ~ 16:30 UTC-3

## Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

### Emilio Lauret

#### Universidad Nacional del Sur, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given $G$ a compact Lie group, we estimate the first Laplace eigenvalue and the diameter of a left-invariant metric on $G$ in terms of its {\it metric eigenvalues}, that is, the eigenvalues of the corresponding positive definite symmetric matrix (w.r.t.\ a fixed bi-invariant metric) associated to a left-invariant metric.

As a consequence, we give a partial answer to the following conjecture by Eldredge, Gordina, and Saloff-Coste [GAFA {\bf 28}, 1321--1367 (2018)]: there exists a positive real number $C$ depending only on $G$ such that the product between the first Laplace eigenvalue and the square of the diameter is bounded by above by $C$ for every left-invariant metric.

The talk is based on the article \url{https://arxiv.org/abs/2004.00350}.

Friday, July 23, 16:40 ~ 17:10 UTC-3

## Low-dimensional double disk bundles

### Fernando Galaz-García

#### Durham University, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

Manifolds that admit a double disk bundle decomposition have played an important role in the study of closed Riemannian manifolds with positive or nonnegative (sectional) curvature. In particular, the presence of such a decomposition has proven useful both in producing new examples and in proving classification results under additional symmetry assumptions. More generally, Grove's Double Soul Conjecture asserts that any closed simply connected manifold of nonnegative curvature is the union of two disk bundles, while the Bott Conjecture asserts that any closed simply connected nonnegative must be rationally elliptic. In this talk I will discuss different sources of examples as well as classification results for (rationally elliptic) double disk bundles in low dimensions.

Joint work with Jason DeVito (University of Tennessee at Martin, USA) and Martin Kerin (National University of Ireland Galway, Ireland).

Friday, July 23, 17:20 ~ 17:50 UTC-3

## A diameter gap for isometric quotients of the unit sphere

### Claudio Gorodski

#### University of São Paulo, Department of Mathematics, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will explain our proof of the existence of $\epsilon>0$ such that every quotient of the unit sphere $S^n$ ($n\geq2$) by an isometric group action has diameter zero or at least $\epsilon$. The novelty is the independence of $\epsilon$ from $n$. The classification of finite simple groups is used in the proof.

Joint work with Christian Lange (University of Cologne, Germany), Alexander Lytchak (University of Cologne, Germany) and Ricardo A. E. Mendes (University of Oklahoma, USA).

Friday, July 23, 19:00 ~ 19:30 UTC-3

## Torus representations with connected isotropy groups: Structural results and applications

### Lee Kennard

#### Syracuse University, U.S.A. - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will discuss recent joint work with Michael Wiemeler and Burkhard Wilking on the topology of manifolds admitting torus-invariant Riemannian metrics with positive sectional curvature. I will focus on one elementary but important part of our proofs, an analysis of the structure of representations of tori with the property that all isotropy groups are connected. Such representations arise any time a compact Riemannian manifold has an isometry group of positive dimension. Our analysis includes a splitting theorem and estimates on the minimum codimension of fixed point sets. More recently, we have observed that these torus representations give rise to totally unimodular matrices and hence provide yet another instance of abstract objects called regular matroids occuring in nature.

Joint work with Michael Wiemeler (University of Muenster) and Burkhard Wilking (University of Muenster).

Friday, July 23, 19:40 ~ 20:10 UTC-3

## Torus actions on Alexandrov $4$-spaces

### Jesús Núñez-Zimbrón

#### CIMAT, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

Alexandrov spaces are metric spaces which are not necessarily smooth but admit a notion of "sectional curvature bounded below". In contrast to manifolds, these spaces may have topological or metric singularities. Nevertheless, many typical results from Riemannian geometry hold in this more general setting. In this talk I will speak about an equivariant classification and a partial topological classification of torus actions on orientable Alexandrov spaces of dimension four, generalizing previous work of Orlik and Raymond originally obtained in the case of four-dimensional manifolds.

Joint work with Diego Corro (Universidad Nacional Autónoma de México) and Masoumeh Zarei (University of Augsburg).

Friday, July 23, 20:20 ~ 20:50 UTC-3

## Upper bound on the revised first Betti number and torus stability for RCD spaces

### Raquel Perales

#### IMATE UNAM, Oaxaca , Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

Gromov and Gallot showed in the past century that for a fixed dimension n there exists a positive number $\varepsilon(n)$ so that any $n$-dimensional riemannian manifold satisfying $Ric_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)$ has first Betti number smaller than or equal to $n$. Furthermore, by Cheeger-Colding if the first Betti number equals $n$ then $M$ is bi-Hölder homeomorphic to a flat torus. This part is the corresponding stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension has to be a torus. The proof of Gromov and Cheeger-Colding results rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of $M$ and then study the geometry of the covering. In this talk we will generalize these results to the case of $RCD(K,N)$ spaces, which is the synthetic notion of a riemannian manifold satisfying $Ric \geq K$ and $dim \leq N$. This class of spaces include ricci limit spaces and Alexandrov spaces.

Joint work with I. Mondello and A. Mondino.

## Posters

## Spaces with Bounded Curvature

### Andrés Ahumada Gómez

#### Universidad Nacional Autónoma de México, México - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will discuss metric spaces with upper and lower curvature bounds (in the sense of A. D. Aleksandrov). We will present a riemannian structure developed by V. N. Berestovskii, and some of its properties. We will then discuss the work of I. G. Nikolaev on a regularization process for the previous riemannian structure and on approximations of our spaces by smooth riemannian manifolds. At the end we will discuss a potential extension to the equiariant setting.

## The Alekseevskii Conjecture in 9 and 10 dimensions

### Rohin Berichon

#### University of Queensland, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The study of Einstein Riemannian manifolds is a broad, yet rich field of study. In the case of homogeneous manifolds, Alekseevskii famously conjectured in 1975 that every connected homogeneous Einstein manifold with negative Ricci curvature is diffeomorphic to Euclidean space. Until now, the conjecture was only known up to dimension 8, besides 5 possible exceptions, and in some cases in dimension 10. Our work shows that noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions.

## Generalized complex and paracomplex structures on product manifolds

### Yamile Godoy

#### CIEM - FAMAF (Conicet - Universidad Nacional de Córdoba), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In 2003 Hitchin introduced generalized complex structures. They can be thought of as geometric structures on a smooth manifold interpolating between complex and symplectic structures, since these ones are particular extremal cases.

Given a product manifold $(M, r)$ we define generalized geometric structures on $M$ which interpolate between two geometric structures compatible with $r$. We study the twistor bundles whose smooth sections are these new structures, obtaining the typical fibers as homogeneous spaces of classical groups. Also, we give examples of Lie groups with a left invariant product structure which admit some of these new structures.

-E. A. Fernández-Culma, Y. Godoy, M. Salvai, Generalized complex and paracomplex structures on product manifolds. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 3, (2020) Paper No. 154.

Joint work with Edison Fernández-Culma (CIEM - FAMAF, Argentina) and Marcos Salvai (CIEM - FAMAF, Argentina).

## Gradient Ambient Obstruction Solitons on Homogeneous Manifolds

### Erin Griffin

#### Seattle Pacific University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. In doing so, we establish a number of results for solitons to the geometric flow by a general tensor $q$.

Focusing on dimension n=4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat; that the only homogeneous, non-Bach-flat, shrinking gradient solitons are product metrics on $\mathbb{R}^2×S^2$ and $\mathbb{R}^2×H^2$; and there is a homogeneous, non-Bach-flat, expanding gradient Bach soliton.

## On Ricci negative derivations

### Valeria Gutiérrez

#### Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. It has been conjectured by Lauret- Will that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimension $≤5$, as well as for Heisenberg and standard filiform Lie algebras.

## Harmonic unit normal sections of the Grassmannian

### Ruth Paola Moas

#### Universidad Nacional de Córdoba - Universidad Nacional de Río Cuarto, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G\left( k,n\right) $ be the Grassmannian of oriented subspaces of $% \mathbb{R}^{n}$ of dimension $k$ with its canonical symmetric Riemannian metric. We study the energy of maps assigning a unit vector normal to $P$ to each $P\in G\left( k,n\right) $. They are sections of a sphere bundle $% E_{k,n}^{1}$ over $G\left( k,n\right) $. The octonionic double and triple cross products induce in a natural way such sections for $k=2,n=7$ and $% k=3,n=8$, respectively. We prove that they are harmonic maps into $% E_{k,n}^{1}$ endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. We also show that these sphere bundles do not have parallel sections, which trivially would have had minimum energy.

In a second intance we analyze the energy of maps assigning an orthogonal complex structure $J\left( P\right) $ on $P^{\bot }$ to each $P\in G\left( 2,8\right) $. They are sections of the unit sphere bundle over $G\left( 2,8\right) $ whose fiber at each $P$ consists essentially of the skew-symmetric transformations on $P^{\bot }$. We prove that the section naturally induced by the octonionic triple product is a harmonic map. We comment on the relationship with the harmonicity of the canonical almost complex structure of $S^{6}$.

Joint work with Francisco Ferraris (Universidad Nacional de Córdoba, Argentina) and Marcos Salvai (Conicet - Universidad Nacional de Córdoba, Argentina).

## Static almost-Kähler structures on Lie groups

### Camilla Molina

#### Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

The symplectic curvature flow is a specially sophisticated equation that evolves almost-Kähler manifolds. The fixed points of this flow, which are analogous to the Einstein metrics for the Ricci flow, are called static structures. Streets and Tian, who first introduced the flow, proved that in dimension $4$ every compact smooth manifold with a static structure is Kähler-Einstein. We show that this rigidity condition is no longer valid in dimension $6$ and above.

## Ancient solutions of the Ricci flow on homogeneous spaces

### Sammy Sbiti

#### University of Pennsylvania, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss the existence of ancient solutions of the Ricci flow on certain classes of homogeneous spaces. As $t\to-\infty$ these solutions collapse with toral fibers to a homogeneous Einstein metric.

Joint work with Francesco Pediconi (Università degli Studi di Firenze, Italy).

## Rigidity and stability of Einstein metrics on homogeneous spaces

### Paul Schwahn

#### University of Stuttgart, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

The question of rigidity of a given Einstein metric $g$, i.e. whether $g$ can be deformed through a curve $g_t$ of Einstein metrics on the same manifold, is closely related to the stability of $g$ under the Einstein-Hilbert action by the fact that Einstein metrics are critical points of the (normalized) total scalar curvature functional. The stability problem for irreducible compact symmetric spaces has been widely investigated by N. Koiso and settled by recent results, using the theory of harmonic analysis on homogeneous spaces.

I give an overview of the results about rigidity and stability on symmetric spaces and present some novel results about the rigidity and stability of the non-symmetric homogeneous spaces like, for example, the 6-dimensional homogeneous nearly Kähler manifolds.

Joint work with Uwe Semmelmann (University of Stuttgart, Germany).

## Positive Hermitian Curvature Flow on complex Lie groups

### James Stanfield

#### The University of Queensland, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The success of the (Kähler)-Ricci flow sparks a natural desire to seek suitable generalisations to non-Kähler Hermitian geometry. In this presentation, we will focus on one such generalisation. Namely, the Positive Hermitian Curvature Flow introduced by Ustinovskiy as part of a family of Hermitian Curvature Flows originally studied by Streets and Tian. This evolution equation is of interest as it preserves many natural curvature positivity conditions for Hermitian manifolds.

We consider the Positive Hermitian Curvature Flow on the space of left-invariant Hermitian metrics on a complex Lie group G, a large class of Hermitian manifolds which are typically non-Kähler. Specifically, we will study the asymptotic behaviour when G is nilpotent or almost-abelian. Time permitting, we will also discuss results regarding soliton solutions in these settings and the case where G is simple.

## Classification of 6-dimensional almost abelian flat solvmanifolds

### Alejandro Tolcachier

#### FAMAF-UNC, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Solvmanifolds, i.e. compact manifolds obtained as quotients of simply-connected solvable Lie groups by discrete subgroups (called lattices), are an important class of manifolds. Some of these solvmanifolds admit a flat Riemannian metric induced by a flat left invariant metric on the associated Lie group. Flat solvmanifolds are a particular class of compact flat manifolds. Such a solvmanifold is isometric to a compact quotient $\mathbb{R}^n/\Gamma$ for some discrete subgroup $\Gamma$ of the isometries of $\mathbb{R}^n$ and its fundamental group is isomorphic to $\Gamma$. These groups were characterized by the so called three Bieberbach Theorems and consequently they are called Bieberbach groups.

In general, it is difficult to determine whether a solvable Lie group admits lattices or not, which makes difficult the construction of solvmanifolds. This poster will talk about a special class of flat solvmanifolds arisen from almost abelian flat Lie groups, that is, groups of the form $\mathbb{R}\ltimes_{\phi} \mathbb{R}^d$ for certain action $\phi$. An advantage is that for almost abelian Lie groups there is a criterion to determine all its lattices. We will focus in the classification of 6-dimensional almost abelian flat solvmanifolds. In order to do so, we will have to solve the problem of finding the conjugacy classes of certain matrices in $\mathsf{GL}(5,\mathbb{Z})$ with finite order.