论文标题
扭曲的Burnside-Frobenius Theorem和$ r_ \ infty $ -property for Lamplighter-type组
Twisted Burnside-Frobenius Theorem and $R_\infty$-Property for Lamplighter-Type Groups
论文作者
论文摘要
我们证明，受限制的花圈产品$ {\ Mathbb {z} _n \ Mathbin {\ Mathrm {wr}}} \ Mathbb {z}^k} $具有$ r_ \ infty $ -property，i。 e。每个自动形态$φ$都有无限的reidemister $ r（φ）$，在两种情况下：（1）对于任何$ k $，甚至$ n $; （2）对于奇数$ k $和$ n $可除以3。在其余的情况下，有有限的reidemister数字的自动形态，我们证明有限二二维扭曲的燃烧效果 - 弗罗比尼乌斯定理（tbft）：$ r（φ）$等于有限的固定级别的不可用的固定级别的数量$ {[ρ] \ mapsto [ρ\crocdφ]} $。
We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $φ$ has infinite Reidemeister number $R(φ)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(φ)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[ρ]\mapsto[ρ\circφ]}$.
