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Katrin Tent

Publications and source records attributed to Katrin Tent.

At least 19 recordsLinked to original sources
arXiv2026

From the Cherlin-Zilber Conjecture via sharply $2$-transitive groups to the Burnside problem

We review the current state of the Cherlin-Zilber Algebraicity Conjecture on simple groups of finite Morley rank, which states that every such group is the group of $K$-rational points of an algebraic group for some algebraically closed field $K$. We will explain the relevance of sharply 2-transitive groups as a potential source of counterexamples and how the Burnside problem necessarily comes into the picture.

math.LO↗
arXiv2025

On universal-homogeneous hyperbolic graphs and spaces and their isometry groups

The Urysohn space is the unique separable metric space that is universal and homogeneous for finite metric spaces, i.e., it embeds any finite metric space any isometry between finite subspaces extends to an isometry of the whole space. We here consider the existence of a universal-homogeneous hyperbolic space. We show that for $\delta>0$ there is no $\delta$-hyperbolic space which is universal and homogeneous in the above sense for all finite $\delta$-hpyerbolic spaces. We then show that for any $\delta\geq 0$ and any countable class $\mathcal{C}$ of $\delta$-hyperbolic spaces with countably many distinguished $\delta$-closed subspaces there exists a $\delta$-hyperbolic metric space $\mathbb{H}_\mathcal{C}$ such that every $X\in \mathcal{C}$ can be embedded into $\mathbb{H}_\mathcal{C}$ as a $\delta$-closed subspace and any isometry between distinguished $\delta$-closed subspaces extends to an isometry of $\mathbb{H}_\mathcal{C}$. If $\mathcal{C}$ consists of $\delta$-hyperbolic geodesic spaces, then $\mathbb{H}_\mathcal{C}$ contains the quasi-tree of spaces as defined by Bestvina et al.. For $\mathcal{C}_\delta$ the class of all finite $\delta$-hyperbolic spaces with rational distances or the class of finite $\delta$-hyperbolic graphs, the limit $\mathbb{H}_\delta$ is a $\delta$-hyperbolic space (or graph, respectively) universal for all finite $\delta$-hyperbolic spaces with rational distances (or finite $\delta$-hyperbolic graphs) and such that any isometry between $\delta$-closed subspaces extends to an isometry of $\mathbb{H}_\delta$. We show that the isometry group of $\mathbb{H}_\delta$ does not contain elements of bounded displacement and has no dense conjugacy class.

math.MG↗
arXiv2024

Omega-categorical pseudofinite groups

We explore the interplay between omega-categoricity and pseudofiniteness for groups, conjecturing that omega-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of omega-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the (omega-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an omega-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on omega-categorical groups over 50 years.

math.LO↗
arXiv2023

Non-split sharply 2-transitive groups of odd positive characteristic

It is well-known that every sharply 2-transitive group of characteristic 3 splits. Here we construct the first examples of non-split sharply 2-transitive groups in odd positive characteristic $p$, for sufficiently large primes $p$. Furthermore, we show that any group without 2-torsion can be embedded into a non-split sharply 2-transitive group of characteristic $p$ for all sufficiently large primes $p$, yielding $2^{\aleph_0}$ many pairwise non-isomorphic countable non-split sharply 2-transitive groups in any sufficiently large characteristic.

math.GR↗
arXiv2023

The Burnside problem for odd exponents

We show that the free Burnside groups $B(m,n)$ are infinite for $m\geq 2$ and odd $n\geq 557$, the best currently known lower bound for the exponent. The proof uses iterated small cancellation theory where the induction is based on the nesting depth of relators. The main instrument at every step is a new concept of a certification sequence.

math.GR↗
arXiv2021

Simple sharply 2-transitive groups

We construct simple sharply 2-transitive groups. Our result answers an open question of Peter Neumann. In fact, we prove that every sharply 2-transitive group of characteristic 0 embeds into a simple sharply 2-transitive group.

math.GR↗
arXiv2021

Universality vs Genericity and $C_4$-free graphs

We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide several new examples of weak Fra\"iss\'e classes of finite graphs. Finally, we show that the class of all countable $C_4$-free graphs does not contain a generic structure, strengthening a result of A. Hajnal and J. Pach.

math.LO↗
arXiv2021

Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank

We define the notion of mock hyperbolic reflection spaces and use it to study Frobenius groups, in particular in the context of groups of finite Morley rank including the so-called bad groups. We show that connected Frobenius groups of finite Morley rank and odd type with nilpotent complement split or interpret a bad field of characteristic zero. Furthermore, we show that mock hyperbolic reflection spaces of finite Morley rank satisfy certain rank inequalities, implying in particular that any connected Frobenius group of odd type and Morley rank at most ten either splits or is a simple non-split sharply 2-transitive group of characteristic different from 2 and of Morley rank 8 or 10.

math.GR↗
arXiv2021

Defining R and G(R)

We show that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are essentially (nearly always) the double centralizers of corresponding root elements. In very many cases this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. For such groups it then follows that the group G(R) is finitely axiomatizable in the appropriate class of groups provided R is finitely axiomatizable in the corresponding class of rings.

math.GR↗
arXiv2020

On the geometry of sharply 2-transitive groups

We show that the geometry associated to certain non-split sharply 2-transitive groups does not contain a proper projective plane. For a sharply 2-transitive group of finite Morley rank we improve known rank inequalities for this geometry and conclude that a sharply 2-transitive group of Morley rank 6 must be of the form $K\rtimes K^*$ for some algebraically closed field $K$.

math.GR↗
arXiv2019

Finite axiomatizability for profinite groups

A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups. Reasons why certain groups cannot be FA are also discussed.

math.GR↗
arXiv2019

Coarse groups, and the isomorphism problem for oligomorphic groups

Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed subgroups of $S_\infty$. We axiomatise them abstractly as structures with a ternary relation. For appropriate classes of groups, including the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular we can recover an isomorphic copy of~$G$ from $\mathcal M(G)$ in a Borel fashion. A closed subgroup $G$ of $S_\infty$ is called oligomorphic if for each $n$, its natural action on $n$-tuples of natural numbers has only finitely many orbits. We use the concept of a coarse group to show that the isomorphism relation for oligomorphic subgroups of $S_\infty$ is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of $S_\infty$ that are topologically isomorphic to oligomorphic groups.

math.LO↗
arXiv2017

On weak Fraisse limits

Using the natural action of $S_\infty$ we show that a countable hereditary class $\mathcal C$ of finitely generated structures has the joint embedding property (JEP) and the weak amalgamation property (WAP) if and only if there is a structure $M$ whose isomorphism type is comeager in the space of all countable, infinitely generated structures with age in $\mathcal C$. In this case, $M$ is the weak Fra\"iss\'e limit of $\mathcal C$. This applies in particular to countable structures with generic automorphisms and recovers a result by Kechris and Rosendal [Proc. Lond. Math. Soc., 2007].

math.LO↗
arXiv2017

Universal-homogeneous structures are generic

We prove that the Fra\"iss\'e limit of a Fra\"iss\'e class $\mathcal C$ is the (unique) countable structure whose isomorphism type is comeager (with respect to a certain logic topology) in the Baire space of all structures whose age is contained in $\mathcal C$ and which are defined on a fixed countable universe. In particular, the set of groups isomorphic to Hall's universal group is comeager in the space of all countable locally finite groups and the set of fields isomorphic to the algebraic closure of $\mathbb F_p$ is comeager in the space of countable fields of characteristic $p$.

math.LO↗