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Register a new userOn the stabilizers of finite sets of numbers in the R. Thompson group $F$
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We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and only if $U$ consists of rational numbers. We also show that such subgroups are isomorphic surprisingly often. In particular, we prove that if finite sets $Usubset [0,1]$ and $Vsubset [0,1]$ consist of rational numbers which are not finite binary fractions, and $|U|=|V|$, then the stabilizers of $U$ and $V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup $bar F<Homeo([0,1])$ which is the completion of $F$ with respect to what we call the Hamming metric on $F$. Moreover the conjugator can be found in a certain subgroup $F < bar F$ which consists of possibly infinite tree-diagrams with finitely many infinite branches. We also show that the group $F$ is non-amenable.
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We prove that Thompsons group $F$ has a subgroup $H$ such that the conjugacy problem in $H$ is undecidable and the membership problem in $H$ is easily decidable. The subgroup $H$ of $F$ is a closed subgroup of $F$. That is, every function in $F$ which is a piecewise-$H$ function belongs to $H$. Other interesting examples of closed subgroups of $F$ include Jones subgroups $overrightarrow{F}_n$ and Jones $3$-colorable subgroup $mathcal F$. By a recent result of the first author, all maximal subgroups of $F$ of infinite index are closed. In this paper we prove that if $Kleq F$ is finitely generated then the closure of $K$, i.e., the smallest closed subgroup of $F$ which contains $K$, is finitely generated. We also prove that all finitely generated closed subgroups of $F$ are undistorted in $F$. In particular, all finitely generated maximal subgroups of $F$ are undistorted in $F$.
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.
We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
A group $G$ is said to be $frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 in G$, there exists $y in G$ such that $G = langle x_1, y rangle = langle x_2, y rangle$. In other words, $s(G) geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
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