An algorithm that decides translation equivalence in a free group of rank two

Translation equivalence in free groups

Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements $g,h$ in a free group $F$ have the property that for every free isometric action of $F$ on an $\mathbb{R}$-tree $X$ the translation lengths of $g$ and $h$ on $X$ are equal. We give a combinatorial characterization of this phenomenon, called translation equivalence, in terms of Whitehead graphs and exhibit two difference sources of it. The first source of translation equivalence comes …

Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation…

晶格轨道多面体的等价性

Equivalence of Lattice Orbit Polytopes

本文研究了晶格轨道多面体的等价性问题，提出正常化等价关系以简化整数凸优化问题，证明当群为QI群时，核心点在正常化等价下的数量有限。

On Tarski's problem for virtually free groups

We give a complete classification of finitely generated virtually free groups up to $\forall\exists$-elementary equivalence. As a corollary, we give an algorithm that takes as input two finite presentations of virtually free groups, and decides whether these groups have the same $\forall\exists$-theory or not.

Property A, partial translation structures and uniform embeddings in groups

We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C*-algebra C*(T) associated with it which is a subalgebra of the uniform Roe algebra C*_u(X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently group-like, as determined by our invariant, properties of the Roe algebra can be deduced from those of C*(T). We also give a proof of the fact that the uniform Roe algebra of a metric space is a coarse invariant up to Morita equivalence.

Relations between counting functions on free groups and free monoids

We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n \geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of sums of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a graphical algorithm to determine whether two given sums of counting functions are equivalent. In particular, this yields an algorithm to decide whether two sums of Brooks quasimorph…