论文标题
汤普森集团$ f $的组圈方程式
Systems of equations over the group ring of Thompson's group $F$
论文作者
论文摘要
令$ r = k [g] $是$ k $上的集团$ g $的组环。众所周知，如果$ g $是可修正的，则$ r $满足矿石条件：对于任何$ a，b \ in r $ in r $ in r $ in r $ in r $ in r $ in r $，以便$ au = bv $，其中$ u \ u \ ne0 $或$ v \ ne0 $。对于可符合的组而言，对于任何有限的线性方程式系统，在$ r $上都存在非零的解决方案，其中未知数超过方程数。最近，巴索迪证明了相反。由于该定理的结果，基拉克（Kielak）证明r。\，汤普森（Thompson）的$ f $在且仅当它满足矿石状况时才可以正常。 $ f $的不适性问题是一个长期的公开问题。 在本文中，我们证明某些方程式或其系统在$ f $的组环中具有非零的解决方案。我们通过Donnelly提高了一些结果，表明存在有限集$ y \ subset f $与属性$ | ay | <\ frac43 | y | $，其中$ a = \ {x_0，x_1，x_2 \} $。这意味着方程系统的一些结果。我们表明，对于$ f $的组环中的任何元素$ b $，方程$（1-x_0）u = bv $具有非零解决方案。 $ 1-x_1 $而不是$ 1-x_0 $的相应事实保持开放。我们推断出任何$ M \ ge1 $ the System $（1-X_0）u_0 =（1-x_1）u_1 = \ cdots =（1-x_m）u_m $在$ f $的组环中具有非零解决方案。我们还分析了方程$（1-x_0）u =（1-x_1）v $，对其在$ k [f] $中的所有解决方案提供了精确的明确描述。这很重要，因为对于$ x_0 $，$ x_1 $ in $ f $之间的任何群体关系很重要，一个人自然可以分配这样的解决方案。因此，这可以帮助估计发电机之间给定长度的关系数量。
Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. It is known that if $G$ is amenable then $R$ satisfies the Ore condition: for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It is also true for amenable groups that a non-zero solution exists for any finite system of linear equations over $R$, where the number of unknowns exceeds the number of equations. Recently Bartholdi proved the converse. As a consequence of this theorem, Kielak proved that R.\,Thompson's group $F$ is amenable if and only if it satisfies the Ore condition. The amenability problem for $F$ is a long-standing open question. In this paper we prove that some equations or their systems have non-zero solutions in the group rings of $F$. We improve some results by Donnelly showing that there exist finite sets $Y\subset F$ with the property $|AY| < \frac43|Y|$, where $A=\{x_0,x_1,x_2\}$. This implies some result on the systems of equations. We show that for any element $b$ in the group ring of $F$, the equation $(1-x_0)u=bv$ has a non-zero solution. The corresponding fact for $1-x_1$ instead of $1-x_0$ remains open. We deduce that for any $m\ge1$ the system $(1-x_0)u_0=(1-x_1)u_1=\cdots=(1-x_m)u_m$ has nonzero solutions in the group ring of $F$. We also analyze the equation $(1-x_0)u=(1-x_1)v$ giving a precise explicit description of all its solutions in $K[F]$. This is important since to any group relation between $x_0$, $x_1$ in $F$ one can naturally assign such a solution. So this can help to estimate the number of relations of a given length between generators.
