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View article: On the model theory of the Farey graph
On the model theory of the Farey graph Open
We axiomatize the theory of the Farey graph and prove that it is $\omega$-stable of Morley rank $\omega$.
View article: On universal-homogeneous hyperbolic graphs and spaces and their isometry groups
On universal-homogeneous hyperbolic graphs and spaces and their isometry groups Open
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 …
View article: OMEGA-CATEGORICAL PSEUDOFINITE GROUPS
OMEGA-CATEGORICAL PSEUDOFINITE GROUPS Open
We explore the interplay between $\omega $ -categoricity and pseudofiniteness for groups, and we conjecture that $\omega $ -categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p…
View article: Omega-categorical pseudofinite groups
Omega-categorical pseudofinite groups Open
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 clas…
View article: Non-split sharply 2-transitive groups of odd positive characteristic
Non-split sharply 2-transitive groups of odd positive characteristic Open
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 $p$. Further…
View article: Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank
Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank Open
We define the notion of mock hyperbolic reflection spaces and use it to study\nFrobenius groups, in particular in the context of groups of finite Morley rank\nincluding the so-called bad groups. We show that connected Frobenius groups of\n…
View article: On the model theory of open generalized polygons
On the model theory of open generalized polygons Open
We show that for any $n\geq 3$ the theory of open generalized $n$-gons is complete, decidable and strictly stable, yielding a new class of examples in the zoo of stable theories.
View article: The Burnside problem for odd exponents
The Burnside problem for odd exponents Open
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 nes…
View article: Defining $R$ and $G(R)$
Defining $R$ and $G(R)$ Open
We show that for Chevalley groups G(R) of rank at least 2 over an integral domain R each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that R and G(R) are bi-interpretabl…
View article: Simple sharply 2-transitive groups
Simple sharply 2-transitive groups Open
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.
View article: Finite axiomatizability for profinite groups
Finite axiomatizability for profinite groups Open
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 k…
View article: Coarse groups, and the isomorphism problem for oligomorphic groups
Coarse groups, and the isomorphism problem for oligomorphic groups Open
Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup G of [Formula: see text] is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has only finite…
View article: Universality vs Genericity and $C_4$-free graphs
Universality vs Genericity and $C_4$-free graphs Open
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 sev…
View article: Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank
Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank Open
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 fin…
View article: Finite axiomatizability for profinite groups
Finite axiomatizability for profinite groups Open
A group is finitely axiomatizable (FA) in a class 𝒞 if it can be determined up to isomorphism within 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 profini…
View article: On the geometry of sharply 2-transitive groups
On the geometry of sharply 2-transitive groups Open
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 a…
View article: Simplicity of the automorphism groups of ordered homogeneous structures
Simplicity of the automorphism groups of ordered homogeneous structures Open
We define the notions of a free fusion of structures and a weakly stationary independence relation. We apply these notions to prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like th…
View article: Finite axiomatizability for profinite groups I: group theory
Finite axiomatizability for profinite groups I: group theory Open
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 k…
View article: Oligomorphic groups are essentially countable
Oligomorphic groups are essentially countable Open
We study the complexity of the isomorphism relation on classes of closed subgroups of $S_\infty$, the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces.
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View article: THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM
THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM Open
We study the complexity of the topological 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 Borel sp…
View article: On weak Fraisse limits
On weak Fraisse limits Open
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 struct…
View article: Universal-homogeneous structures are generic
Universal-homogeneous structures are generic Open
We prove that the Fraïssé limit of a Fraïssé 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 i…
View article: A sharply 2-transitive group without a non-trivial abelian normal subgroup
A sharply 2-transitive group without a non-trivial abelian normal subgroup Open
We show that any group G is contained in some sharply 2-transitive group \mathcal G without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups \mathcal G that we construct have …
View article: Some model theory of profinite groups
Some model theory of profinite groups Open
We give some background on uniform pro-p groups and the model theory of profinite NIP groups.
View article: An alternative axiomization of $N$-pseudospaces
An alternative axiomization of $N$-pseudospaces Open
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 forki…
View article: \emph{Addendum to} Sharply $2$-transitive groups of characteristic~$0$ [arXiv:1604.00573]
\emph{Addendum to} Sharply $2$-transitive groups of characteristic~$0$ [arXiv:1604.00573] Open
In this short note we show how to modify the construction of non-split sharply $2$-transitive groups of characteristic~$0$ given by Rips and Tent [arXiv:1604.00573] to allow for arbitrary fields of characteristic 0
View article: Sharply 2-transitive groups of characteristic 0
Sharply 2-transitive groups of characteristic 0 Open
We construct sharply 2-transitive groups of characteristic 0 without regular normal subgroups. These groups act sharply 2-transitively by conjugation on their involutions. This answers a long-standing open question.
View article: Building-like geometries of finite Morley Rank
Building-like geometries of finite Morley Rank Open
For any $n\geq 6$ we construct almost strongly minimal geometries of type $\bullet \overset{n}{-} \bullet \overset{n}{-}\bullet$ which are $2$-ample but not $3$-ample.