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Annalisa Conversano

Publications and source records attributed to Annalisa Conversano.

13 recordsLinked to original sources
arXiv2025

A Jordan-Chevalley decomposition beyond algebraic groups

We prove a decomposition of definable groups in o-minimal structures generalizing the Jordan-Chevalley decomposition of linear algebraic groups. It follows that any definable linear group G is a semidirect product of its maximal normal definable torsion-free subgroup N(G) and a definable subgroup P, unique up to conjugacy, definably isomorphic to a semialgebraic group. Along the way, we establish two other fundamental decompositions of classical groups in arbitrary o-minimal structures: 1) a Levi decomposition and 2) a key decomposition of disconnected groups, relying on a generalization of Frattini's argument to the o-minimal setting. In o-minimal structures, together with p-groups, 0-groups play a crucial role. We give a characterization of both classes and show that definable p-groups are solvable, like finite p-groups, but they are not necessarily nilpotent. Furthermore, we prove that definable p-groups (p=0 or p prime) are definably generated by torsion elements and, in definably connected groups, 0-Sylow subgroups coincide with p-Sylow subgroups for each p prime.

math.LO↗
arXiv2025

Ring theory in o-minimal structures

We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an infinite field and it is essentially semialgebraic. A surprisingly strong correspondence between definably connected rings and finite-dimensional associative $\mathbb{R}$-algebras is established. Every ideal of a definable unital ring is definable, from which it follows that every definable unital ring is Artinian and Noetherian. If a definable ring $R$ is not unital, we give necessary and sufficient conditions for $R$ to embed in a definable unital ring as an ideal. Moreover, when this is the case, we provide the smallest such definable unital ring $R^{\wedge}$, its definable unitazation.

math.LO↗
arXiv2023

Lie groups in the symmetric group: reducing Ulam's problem to the simple case

Ulam asked whether all Lie groups can be represented faithfully on a countable set. We establish a reduction of Ulam's problem to the case of simple Lie groups. In particular, we solve the problem for all solvable Lie groups and more generally Lie groups with a linear Levi component. It follows that every amenable locally compact second countable group acts faithfully on a countable set.

math.GR↗
arXiv2023

Definable rank, o-minimal groups, and Wiegold's problem

We show that an o-minimal structure M defines groups with infinite definable rank if and only if M defines some finite power of $\mathbb{Q}/\mathbb{Z}$. If no interval of M is countable, then all groups definable in M have finite definable rank. In general, we prove that every definable group $G$ in an arbitrary o-minimal structure is an extension of a definable periodic group $P$ by a (maximal unique) definably connected definably finitely generated subgroup $\widehat G$. When $G$ is definably connected, $P$ is abelian and the extension almost split, in that $G$ is an almost direct product $G = (\widehat G \times P)/F$, for some finite central subgroup $F$. The definable rank of $\widehat G$ is bounded above by its dimension, and the upper bound is strict whenever $\widehat G$ is not solvable. Along the way, we show that every linear definable group has finite definable rank. This provides another proof, and a generalization to linear o-minimal groups, of the fact that linear algebraic groups over an algebraically closed field of characteristic $0$ contain a Zariski-dense finitely generated subgroup. We further prove that every perfect definable group is normally monogenic, generalizing the finite group case. This yields a positive answer to Wiegold's problem in the o-minimal setting.

math.LO↗
arXiv2021

One Lie group to define them all

We produce a connected real Lie group that, as a first order structure in the group language, interprets the real field expanded with a predicate for the integers. Moreover, the domain of our interpretation is definable in the group.

math.LO↗
arXiv2020

Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.

math.LO↗
arXiv2011

On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.

math.LO↗
arXiv2011

Connected components of definable groups, and o-minimality II

We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.

math.LO↗
arXiv2011

Connected components of definable groups and o-minimality I

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin, Newelski, Petrykovski. The examples also give new non G-compact first order theories.

math.LO↗
arXiv2009

Lie-like decompositions of groups definable in o-minimal structures

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.

math.LO↗