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Katrin Tent

Publications and source records attributed to Katrin Tent.

46 records · Page 3Linked to original sources
arXiv2012

Pseudofinite groups with NIP theory and definability in finite simple groups

We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group which is not soluble-by-finite. However, if C is a class of finite groups such that all infinite ultraproducts of members of C have NIP theory, then there is a bound on the index of the soluble radical of any member of C. We also survey some ways in which model theory gives information on families of finite simple groups, particularly concerning products of images of word maps.

math.LO↗
arXiv2012

On the isometry group of the Urysohn space

We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.

math.GR↗
arXiv2011

The free pseudospace is n-ample, but not (n+1)-ample

We give a uniform construction of free pseudospaces of dimension n extending work by Baudisch and Pillay. This yields examples of $ω$-stable theories which are n-ample, but not (n+1)-ample. The prime models of these theories are buildings associated to certain right-angled Coxeter groups.

math.LO↗
arXiv2011

Strongly transitive multiple trees

We give an amalgamation construction of free multiple trees with a strongly transitive automorphism group. The construction shows that any partial codistance function on a tuple of finite trees can be extended to yield multiple trees.

math.GR↗
arXiv2010

Low functions of reals

We introduce a new notion of computable function on $\R^N$ and prove some basic properties. We give two applications, first a short proof of Yoshinaga's theorem that periods are \el (they are actually low). We also show that the low complex numbers form a algebraically closed field closed under exponentiation and some other special functions.

math.LO↗
arXiv2009

A Maslov cocycle for unitary groups

We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class.

math.KT↗
arXiv2004

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) is at least 2, and let $Γ$ be a uniform lattice in G. (a) If $CH$ holds, then $Γ$ has a unique asymptotic cone up to homeomorphism. (b) If $CH$ fails, then $Γ$ has $2^{2^ω}$ asymptotic cones up to homeomorphism.

math.GT↗
arXiv2004

Asymptotic cones and ultrapowers of Lie groups

Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the 'large-scale structure' of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.

math.GT↗
arXiv2002

Affine $Λ$-buildings, ultrapowers of Lie groups and Riemannian symmetric spaces: an algebraic proof of the Margulis conjecture

In this paper, we give a general group-theoretic construction of affine $\RR$-buildings, and more generally, of affine $Λ$-buildings, associated to semisimple Lie groups over nonarchimedean real closed fields. The construction of Kleiner-Leeb using the asymptotic cone of a Riemannian symmetric space appears as a special case. The explicit knowledge of the building arising here as the asymptotic cone simplifies the proof of the Margulis conjecture due to Kleiner-Leeb.

math.DG↗