arXiv subjects

Gili Golan

Publications and source records attributed to Gili Golan.

23 records · Page 2Linked to original sources

arXiv2015

On Jones' subgroup of R. Thompson group $F$

Recently Vaughan Jones showed that the R. Thompson group $F$ encodes in a natural way all knots, and a certain subgroup $\vec F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\vec F$. In particular we prove that the subgroup $\vec F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i, i=0,1,2,...$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\vec F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\vec F$ is irreducible.

math.GR↗

arXiv2014

The Tarski numbers of groups

The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. In this paper we investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and known properties of Golod-Shafarevich groups, we show that the Tarski numbers of 2-generated non-amenable groups can be arbitrarily large. We also use the cost of group actions to show that there exist groups with Tarski numbers 5 and 6. These provide the first examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely.

math.GR↗

arXiv2014

Tarski numbers of group actions

The Tarski number of an action of a group G on a set X is the minimal number of pieces in a paradoxical decomposition of it. For any k>3 we construct a faithful transitive action of a free group of rank k-1 with Tarski number k. Using similar techniques we construct a group action of a free group F with Tarski number 6 such that the Tarski numbers of restrictions of this action to finite index subgroups of F are arbitrarily large.

math.GR↗

arXiv2014

Groups with Tarski number 5

The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. Until now the only numbers which were known to be Tarski numbers of some groups were 4 and 6. We construct a group with Tarski number 5 and mention a related result for Tarski numbers of group actions.

math.GR↗

arXiv2013

Hindman's Coloring Theorem in arbitrary semigroups

Hindman's Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers $a_1,a_2,\dots$ such that all of the sums $a_{i_1}+a_{i_2}+\dots+a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same color. The celebrated Galvin--Glazer proof of Hindman's Theorem and a classification of semigroups due to Shevrin, imply together that, for each finite coloring of each infinite semigroup $S$, there are distinct elements $a_1,a_2,\dots$ of $S$ such that all but finitely many of the products $a_{i_1}a_{i_2}\cdots a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same color. Using these methods, we characterize the semigroups $S$ such that, for each finite coloring of $S$, there is an infinite \emph{subsemigroup} $T$ of $S$, such that all but finitely many members of $T$ have the same color. Our characterization connects our study to a classical problem of Milliken, Burnside groups and Tarski Monsters. We also present an application of Ramsey's graph-coloring theorem to Shevrin's theory.

math.CO↗