The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups
The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups
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DOI：
10.1093/qmath/hax012
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发表时间：
2015-07
期刊：
影响因子：
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通讯作者：
Susumu Hirose;E. Kin
中科院分区：
文献类型：
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作者：
Susumu Hirose;E. Kin
考虑了属$g$闭曲面上的超椭圆柄体群。这是$g$属的闭曲面上映射类群的子群，由曲面上具有固定超椭圆对合交换并扩展到柄体上的同胚的同位素类组成。证明了$g$属的超椭圆柄体群中所有伪anosov元的最小膨胀（即最小熵）的对数与$1/g$相当。这意味着所讨论的格$g$子群的最小伪anosov扩张的渐近行为与格$g$的环境映射类群的渐近行为相同。我们还确定了超椭圆柄体群的有限表示形式。
We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the subgroup of genus $g$ in question is the same as that of the ambient mapping class group of genus $g$. We also determine finite presentations of the hyperelliptic handlebody groups.