Researches on the spectrum of critical exponents of normal subgroups and the rigidity of cogrowth for hyperbolic discrete groups

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K13767
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Waseda University
• Principal Investigator: Matsuzaki Katsuhiko 早稲田大学, 教育・総合科学学術院, 教授 (80222298)
• Co-Investigator(Kenkyū-buntansha): イェーリッシュ ヨハネス 島根大学, 学術研究院理工学系, 講師 (90741869)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 幾何学的群論 / 収束指数 / ラプラシアンのスペクトル / 自由群 / ケーリーグラフ / 群不変等角測度 / 群上のランダムウォーク

Outline of Final Research Achievements

The Grigorchuk cogrowth formula gave a relationship between the exponent of convergence for the isometric action of normal subgroups on the Cayley graph of the free group and the bottom of the spectrum of the discrete Laplacian on the quotient graph. Similar results were proved by Sullivan for Kleinian groups acting on the hyperbolic space and for the Laplacian on the hyperbolic manifold. A common phenomenon is that the phase transition of the exponents occurs at 1/2 of that of the base group. In this study, even when the lengths of edges of the Cayley graph of a free group is varied, a weighted discrete Laplacian depending on the exponent of convergence of a normal subgroup determines the spectrum, and the generalization of the cogrowth formula was proved for this. The phase transition at 1/2 of the exponent was also verified in this case.

Free Research Field

双曲幾何学

Academic Significance and Societal Importance of the Research Achievements

グラフ上の離散ラプラシアンやグラフの指数増大度の研究は多方面で数多くの研究があるが，グラフの各辺に与える重みの解釈によって，それはネットワーク上のマルコフ連鎖の研究とみることもできれば，双曲性をもつ距離グラフの研究としてみることもできる．本研究の成果は，その両者の立場の間の関係を記述したものと捉えることができる．このような無限グラフの構造をもつネットワークの理論は，情報社会や人間社会の複雑で多様なネットワークの解析に応用をもつことが期待される．