On continuity of growth rates and spectral radii on the space of marked groups

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14318
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Ashikaga University (2023); Waseda University (2020-2022)
• Principal Investigator: Yukita Tomoshige 足利大学, 工学部, 講師 (80843903)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: コクセター系 / 増大度 / Salem数 / Pisot数 / Perron数 / 双曲幾何 / 双曲多面体

Outline of Final Research Achievements

We considered the set of Coxeter systems with N generators as a subspace of the space of marked groups. Then we showed that the space of Coxeter systems is compact and the growth rates are continuous as a function on the space. As an application of this result, we studied arithmetic nature of the growth rates of Coxeter systems. We showed that if the Euler characteristic of the nerve of a 2-dimensional Coxeter system is positive (resp. zero), then the growth rate is a Salem number (resp. Pisot number). These results are extensions of the previous results on the growth rates of discrete hyperbolic reflection groups.

Free Research Field

幾何学的群論

Academic Significance and Societal Importance of the Research Achievements

有限生成群の増大度関数や増大度についての研究は、MilnorのRiemann多様体と基本群の関係についての研究から始まり、Gromovらにより行われた。特に、多項式増大を持つ群は有限指数のベキ零部分群をもつというGromovの多項式増大定理は好例である。本研究では、コクセター系に焦点を当てて増大度についての研究を行い、特に標識付き群を変数とする関数としての連続性や数論的性質について、双曲幾何で得られてきた結果をコクセター系に一般化したものである。