Journées en mémoire d'Eric Jaligot
Program
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Thursday |
Friday |
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9:30 - Welcome coffee |
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10 - Cherlin |
10 - Sela |
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11:10 - Macpherson |
11:10 - Deloro |
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Lunch Break |
Lunch Break |
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14 - Frecon |
14 - Baro |
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14:40 - Khelif |
14:40 - Perin |
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15:30 - Poizat |
15:30 - Borovik |
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16 - Neman |
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Abstracts
E. Baro, And Carter became Cartan
In this talk I will survey the work that Eric Jaligot did together with Margarita Otero and me during his visit to Madrid in the period 2009-2010.
Carter subgroups of groups of finite Morley rank were extensively studied by several authors including Jaligot.
The main open conjecture in this subject is the conjugacy of the Carter subgroups of a group of FMR (true in the solvable case).
The original plan was to study the situation in a group definable in an o-minimal structure.
The first obstacle we found was the definability of the commutator subgroup of a group definable in an o-minimal structure.
Over a year before, Conversano showed an example of such a group with a non-definable commutator subgroup. In [1] we proved
that Conversano's example is essentially the unique obstruction to the definability of the commutator subgroup in the o-minimal context.
In particular, we deduced that if \(G\) is a connected solvable group definable in an o-minimal structure then \(G'\) is definable and connected.
With this result at hand, in [2] we embarked on the study of Carter subgroups in the o-minimal setting.
In the solvable case, we were able to show the conjugacy of Carter subgroups
(as it happens in the FMR context) but we quickly realized that the conjecture was false in general.
However, this led us to the concept of Cartan subgroups and their development in the o-minimal context.
[1] E. Baro, E. Jaligot, M. Otero, Commutators in groups definable in o-minimal structures, Proc. AMS 140 (2012), no. 10, 3629--3643.
[2] E. Baro, E. Jaligot, M. Otero, Cartan subgroups of groups definable in o-minimal structures,
to appear in J. Inst. Math. Jussieu.
A. Borovik , Reification of involutions in black box groups
G. Cherlin, Generix begins from mixed punchlines to mikshakes
A. Deloro, Locally soluble groups of finite Morley rank
O. Frécon,
Elementary equivalence of algebraic groups and expanded pure
groups
We consider two algebraic groups \(G\) and \(H\) defined over the field
\(\overline{\mathbb Q}\), and the groups of rational points \(G(K)\) and \(H(K)\) where \(K\) is an algebraically
closed field of characteristic zero. We show that if the pure groups \(G(K)\) and
\(H(K)\) are elementarily equivalent, then they are isomorphic as abstract groups.
The proof requires the analysis of the following minimal group:
\[ \left \{ \begin{pmatrix} t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{pmatrix} \big| \ t \in K^*, \ (a,u,v) \in K^3 \ \right\}
\]
In this centerless algebraic group, the maximal tori are not definable in the
pure group; they are invariant under the abstract group automorphisms when
\(K \cong \overline{\mathbb Q}\), and not generally. The analysis of \(G_{crit}\) leads us to the existence of
a natural category of groups, intermediate between the category of abstract
groups and the category of algebraic groups: the expanded pure groups.
By working with expanded pure groups, we may analyze the group structure
of algebraic groups over \(Q\) and prove our theorem.
The main theorem uses Carter subgroups of groups of finite Morley rank,
whose existence in any group of finite Morley rank was established in a joint
work with Eric Jaligot.
This talk will be in French with slides in English.
A. Khelif, Le tournoi aléatoire comme tournoi de Cayley
D. Macpherson,
Groups definable in valued fields
A. Neman
C. Perin Free products of homogeneous groups
B. Poizat , Jaligot
Z. Sela , Equations over Groups and Semigroups