Groupes fondamentaux des variétés algébriques complexes lisses // Fundamental groups of smooth algebraic complex manifolds

There exists infinite simple finitely presented groups. A positive answer to an open question posed by Serre 70 years ago would imply they should not appear as fundamental groups of complex algebraic manifolds.

Serre a posé il y a 70 ans le probleme de caracteriser les groupes fondamentaux de varietes algebriques complexes lisses parmi les groupes de presentation finie.

J'ai plusieurs problemes tournat autour de cette question qui peuvent etre regardés par un thésard

0) Lien entre groupes de Mumford Tate et groupes de Kahler nilpotents.

1) Il existe des groupes de presentation finie infinis et simples. Une question ouverte posée exclue qu'ils soient des groupes fondamentaux de varoetes algebriques lisses. par Serre. Deux classes ont ete exclues comme groupes de Kahler: les groupes deThompson groups et ceux de Burger-Mozes .

Peut on les exclure comme groupes quasi kahleriens?

2) Une autre classe, construite par Caprace-Remy sont des groupes de Kac Moody agisaant sur un immeucle hyperbolique.
Avant de seulement d'essayer de les exclure comme groupes de Kahler, se pose la question si on peut prouver un theoreme de regularité de Gromov-Schoen pour l'application harmonique equivariante attachee a un morphisme depuis le groupe fondamental d'une surface de Riemann.

Si on peut, une variante en dimension spuperieure permettra des les exclure.

3) Higman et d'autres ont construit d'autres classes encore. Il serait interessant de les explorer systemeqtiquement du point de vue du probleme de Serre.

4) Prouver ou infirmer la conjecture de Shafarevich pour les compactifications toroidales de petit volume des quotients de la boule.

Liste ordonnée selon mon estimation de la difficulté.
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Serre posed 70 years ago the problem of characterizing the fundamental groups of complex algebraic manifolds as finitely presented groups.

I have many problems revolving around this question that can be investigated by a phd student

0) Mumford Tate groups and nilpotent Kahler groups.

1) There exists infinite simple finitely presented groups. A positive answer to an open question posed by Serre 70 years ago would imply they should not appear as fundamental groups of complex algebraic manifolds.

Two classes have been excluded as kahler groups: Thompson groups and Burger-Mozes groups.
Can one exclude them as quasi kahler groups?

2) One other class constructed by Caprace-Remy are Kac Moody groups acting on a hyperbolic building.

Before even trying to exclude them as kahler groups: can one prove a Gromov-Schoen regularity theorem for the harmonic map attached to a morphism from the pi1 of a compact Riemann surface to such a group?

If one can, then a higher dimensionnal GS regularity theorem would exclude these groups as kahler groups.

3) there are other classes constructed by Higman and many group theorists. Explore them systematically and exclude them as kahler groups.

4) explore the Shafarevich Conjecture for small volume toroidal compactifications of ball quotients. I don't know how to exclude them as counter examples.

Problems 0) -4) are ordered according to my estimation of the difficulty.
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Début de la thèse : 01/10/2026

Concours allocations

Université Grenoble Alpes

217 MSTII - Mathématiques, Sciences et technologies de l'information, Informatique

Connaissances avancées en geometrie complexe, bases en geometrie algebrique
Advanced knowledge on Complex Geometry, basic algebraic geometry