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Todor Tsankov

Publications and source records attributed to Todor Tsankov.

26 records · Page 2Linked to original sources
arXiv2015

Free actions of free groups on countable structures and property (T)

We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such $G$ include automorphism groups of countable $ω$-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of $G$ acting freely in all infinite transitive permutation representations of $G$.

math.GR↗
arXiv2015

On the complexity of topological conjugacy of Toeplitz subshifts

In this paper, we study the descriptive set theoretic complexity of the equivalence relation of conjugacy of Toeplitz subshifts of a residually finite group $G$. On the one hand, we show that if $G = \mathbb{Z}$, then topological conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast, if $G$ is non-amenable, then conjugacy of Toeplitz $G$-subshifts is a non-amenable equivalence relation. The results were motivated by a general question, asked by Gao, Jackson and Seward, about the complexity of conjugacy for minimal, free subshifts of countable groups.

math.LO↗
arXiv2014

Extremely amenable groups via continuous logic

We establish a characterization of extreme amenability of any Polish group in Fraïssé-theoretic terms in the setting of continuous logic, mirroring a theorem due to Kechris, Pestov and Todorcevic for closed subgroups of the permutation group of an infinite countable set.

math.LO↗
arXiv2012

Generic representations of abelian groups and extreme amenability

If $G$ is a Polish group and $Γ$ is a countable group, denote by $\Hom(Γ, G)$ the space of all homomorphisms $Γ\to G$. We study properties of the group $\cl{π(Γ)}$ for the generic $π\in \Hom(Γ, G)$, when $Γ$ is abelian and $G$ is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on $Γ$, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $\cl{π(Γ)}$; in the other two, we show that the generic $\cl{π(Γ)}$ is extremely amenable. We also show that if $Γ$ is torsion-free, the centralizer of the generic $π$ is as small as possible, extending a result of King from ergodic theory.

math.LO↗
arXiv2012

Unitary representations of oligomorphic groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group $S_\infty$, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand--Raikov theorem holds for topological subgroups of $S_\infty$: for all such groups, continuous irreducible representations separate points in the group.

math.GR↗
arXiv2012

Decidability of definability

For a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ_0,ϕ_1,...,ϕ_n that define relations R_0,R_1,...,R_n over Γ. The question is whether the relation R_0 is primitive positive definable from R_1,...,R_n, i.e., definable by a first-order formula that uses only relation symbols for R_1,..., R_n, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden). We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

math.LO↗
arXiv2009

The additive group of the rationals does not have an automatic presentation

We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are $p$-divisible for infinitely many primes $p$, or $Q/Z$. The proof is combinatorial and uses most notably Freiman's theorem on sets with small doubling.

math.LO↗