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Title:Rigidity in free groups
Author(s):Ray, Brian
Director of Research:Kapovitch, Ilia
Doctoral Committee Chair(s):Leininger, Christopher J.
Doctoral Committee Member(s):Kapovitch, Ilia; Schupp, Paul E.; Athreya, Jayadev S.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Free Groups
Spectral Rigidity
Geodesic Currents
Abstract:Culler-Vogtmann Outer Space (denoted $\cv_N$) is the space of all free minimal discrete isometric actions of the free group of rank $N$ ($F_N$) on $\mathbb{R}$-trees, $T$. We say a subset $\Sigma \subseteq F_N$ is \emph{spectrally rigid} (resp. \emph{strongly spectrally rigid}) if whenever $T_1, T_2 \in \cv_N$ (resp. the closure, $\overline{\cv}_N$, of $\cv_N$) are $\mathbb{R}$-trees for which $\| \sigma \|_{T_1} = \| \sigma \|_{T_2}$ for every $\sigma \in \Sigma$, then $T_1 = T_2$ in $\cv_N$ (resp. $\overline{\cv}_N$). Similarly, given $T \in \cv_N$ (resp. $\overline{\cv}_N$), we say that $\Sigma \subset F_N$ is \emph{relatively rigid} (resp. \emph{strongly relatively rigid}) for $T$ if whenever $T' \in \cv_N$ (resp. $\overline{\cv}_N$) is such that $\| \sigma \|_T = \| \sigma \|_{T'}$ for every $\sigma \in \Sigma$, then $T = T'$ in $\cv_N$ (resp. $\overline{\cv}_N$). We say that $S \subset \Curr(F_N)$ is a \emph{rigid set of currents} if whenever $T_1, T_2 \in \cv_N$ are such that $\langle \sigma , T_1 \rangle = \langle \sigma , T_2 \rangle$ for every $\sigma \in \Sigma$, then $T_1 = T_2$ in $\cv_N$. The general theory of (non-abelian) actions of groups on $\mathbb{R}$-trees establishes that $T \in \cv_N$ is uniquely determined by its translation length function $\| \cdot \|_T \colon F_N \to \mathbb{R}$, and consequently that $F_N$ itself is spectrally rigid. Results of Smillie and Vogtmann, and of Cohen, Lustig, and Steiner establish that no finite $\Sigma$ is spectrally rigid. Capitalizing on their constructions, we prove that for any $\Phi \in \Aut(F_N)$ and $g \in F_N$, the set $\Sigma = \{ \Phi^n(g) \}_{n \in \mathbb{Z}}$ is not spectrally rigid. We also prove that if $\{ H_i \}_{i=1}^k$ is a finite collection of subgroups, each of infinite index, and $g_i \in F_N$, then $\cup_{i=1}^k g_i H_i$ is not spectrally rigid in $F_N$. Taking $H_i = 1$, we recover the results about finite sets. We also prove that any coset of a nontrivial normal subgroup $H \lhd F_N$ is spectrally rigid. Carette, Francaviglia, Kapovich, and Martino prove that every $T \in \cv_N$ admits a finite relatively rigid set. We prove that this set actually affords strong relative rigidity. We also prove nonexistence of finite strongly relatively rigid sets for certain classes of trees on the boundary, $\partial \cv_N$, of $\cv_N$. We also show that in volume-normalized outer space certain simplicial trees `almost' admit finite strongly relatively rigid sets. Lastly, we prove that arational free trees admit finite strongly relatively rigid sets of currents.
Issue Date:2014-05-30
Rights Information:Copyright 2014 Brian Ray
Date Available in IDEALS:2014-05-30
Date Deposited:2014-05

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