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Katrin Tent

Publications and source records attributed to Katrin Tent.

At least 37 records · Page 2Linked to original sources
arXiv2018

The complexity of topological group isomorphism

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound, which is not sharp because the relation is Borel.

math.LO↗
arXiv2017

An alternative axiomization of $N$-pseudospaces

We give a new axiomatization of the N-pseudospace, studied in [2] (Tent(2014)) and [1] (Baudisch,Martin-Pizarro,Ziegler(2014)) based on the zigzags introduced in [2]. We also present a more detailed account of the characterization of forking given in [2].

math.LO↗
arXiv2016

Describing finite groups by short first-order sentences

We say that a class of finite structures for a finite first-order signature is $r$-compressible if each structure $G$ in the class has a first-order description of size at most $O(r(|G|))$. We show that the class of finite simple groups is $\log$-compressible, and the class of all finite groups is $\log^3$-compressible. As a corollary we obtain that the class of all finite transitive permutation groups is $\log^3$-compressible. The result relies on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators, and group cohomology. We also indicate why the results are close to optimal.

math.LO↗
arXiv2016

Sharply 2-transitive groups in characteristic 0

We construct sharply 2-transitive groups of characteristic 0 without non-trivial abelian normal subgroup. These groups act sharply 2-trnaisitvely by conjugation on their involutions. This answers a longstanding open question.

math.GR↗
arXiv2016

Profinite groups with NIP theory and $p$-adic analytic groups

We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of {\em all} open subgroups, then the first order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index which is a direct product of finitely many compact $p$-adic analytic groups, for distinct primes $p$. In fact, the condition NIP can here be weakened to NTP${}_2$. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup.

math.LO↗
arXiv2016

Infinite sharply multiply transitive groups

The finite sharply $2$-transitive groups were classified by Zassenhaus in the 1930's. They essentially all look like the group of affine linear transformations $x\mapsto ax+b$ for some field (or at least near-field) $K$. However, the question remained open whether the same is true for infinite sharply $2$-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus' results might extend might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal it was indeed proved that all examples inside the given class arise in this way as affine groups. However, it recently turned out that the reason for the lack of a general proof was the fact that there are plenty of sharply $2$-transitive groups which do not arise from fields or near-fields! In fact, it is not too hard to construct concrete examples (see below). In this note, we survey general sharply $n$-transitive groups and describe how to construct examples not arising from fields.

math.GR↗
arXiv2015

A non-desarguesian projective plane

We construct a new non-desarguesian projective plane from a complex analytic structure. At the same time the construction can be explained in terms of so called Hrushovski's construction. This supports the hypothesis that in general structures produced by Hrushovski's construction have "prototypes" in complex geometry.

math.LO↗
arXiv2015

Simplicity of the automorphism groups of some Hrushovski constructions

We show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the 'uncollapsed' structures of infinite Morley rank obtained by the ab initio construction and the (unstable) omega-categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of Lascar and of Tent and Ziegler.

math.LO↗
arXiv2015

Sharply 3-transitive groups

We construct the first sharply $3$-transitive groups not arising from a near field, i.e. point stabilizers have no nontrivial abelian normal subgroup.

math.GR↗
arXiv2014

Sharply $2$-transitive groups

We give an explicit construction of sharply $2$-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.

math.GR↗
arXiv2013

New simple groups with a BN-pair

We show that there are simple groups with a spherical BN-pair of rank 2 which are non-Moufang and hence not of algebraic origin.

math.GR↗
arXiv2013

On groups and fields interpretable in torsion-free hyperbolic groups

We prove that the generic type of a non-cyclic torsion-free hyperbolic group G is foreign to any interpretable abelian group, hence also to any interpretable field. This result depends, among other things, on the definable simplicity of a non-cyclic torsion-free hyperbolic group, and we take the opportunity to give a proof of the latter using Sela's description of imaginaries in torsion-free hyperbolic groups. We also use the description of imaginaries to prove that if F is a free group of rank > 2 then no orbit of a finite tuple from F under Aut(F) is definable.

math.LO↗
arXiv2012

Ampleness in the free group

We show that the theory of the free group -- and more generally the theory of any torsion-free hyperbolic group -- is $n$-ample for any $n\geq 1$. We give also an explicit description of the imaginary algebraic closure in free groups.

math.GR↗