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Fundamental groups and group presentations with bounded relator lengths

Sergio Zamora

Abstract

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group $G$ acts by isometries on a compact geodesic space $X$ whose first Betti number vanishes, then diam$(X) / $diam$(X / G ) \leq 4 \sqrt{ \vert G \vert }$. For a group $G$ and a finite symmetric generating set $S$, $P_k(Γ(G, S))$ denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph $Γ$ of $G$ with respect to $S$ and whose 2-cells are $m$-gons for $0 \leq m \leq k$, defined by the simple graph loops of length $m$ in $Γ$, up to cyclic permutations. Let $G$ be a finite abelian group with $\vert G \vert \geq 3$ and $S$ a symmetric set of generators for which $P_k(Γ(G,S))$ has trivial first Betti number. We show that the first nontrivial eigenvalue $-λ_1$ of the Laplacian on the Cayley graph satisfies $λ_1 \geq 2 - 2 \cos ( 2 π/ k ) $. We also give an explicit upper bound on the diameter of the Cayley graph of $G$ with respect to $S$ of the form $O (k^2 \vert S \vert \log \vert G \vert )$. Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair $(G,S)$ are also obtained.

Fundamental groups and group presentations with bounded relator lengths

Abstract

We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group acts by isometries on a compact geodesic space whose first Betti number vanishes, then diamdiam. For a group and a finite symmetric generating set , denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph of with respect to and whose 2-cells are -gons for , defined by the simple graph loops of length in , up to cyclic permutations. Let be a finite abelian group with and a symmetric set of generators for which has trivial first Betti number. We show that the first nontrivial eigenvalue of the Laplacian on the Cayley graph satisfies . We also give an explicit upper bound on the diameter of the Cayley graph of with respect to of the form . Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair are also obtained.

Paper Structure

This paper contains 17 sections, 22 theorems, 94 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Diameter
  3. Kazhdan constant, Cheeger constant, and spectral gap
  4. Outline
  5. Examples and further problems
  6. Diameter
  7. Spectral gap
  8. (Lack of) Gromov--Hausdorff precompactness
  9. Preliminaries
  10. Notation
  11. Graphs and CW-complexes
  12. Constructing covering spaces
  13. Representation theory of finite groups
  14. Kazhdan constant, Cheeger constant, and spectral gap
  15. Random walks on Cayley graphs
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Let $X$ be a compact geodesic space and $G \leq {\text{Iso}} (X)$ a finite group of isometries. If the first Betti number ${\text{b}_1} (X)$ vanishes, then

Figures (5)

  • Figure 1: We can use the knowledge of the angles in the triangle $OF_1P$ to deduce the length of the segment $OP$. We then proceed to compute the length of the segment $OV_1$ using the length of the segment $OP$ and the known angles of the triangle $OPV_1$.
  • Figure 2: Basic trigonometry shows that if the angle between the unit vectors $x$ and $\rho (s) x$ is $\theta$, then the distance between the endpoints is $2 \sin (\theta / 2)$.
  • Figure 3: If $\omega = s_1 \cdots s_m$, then the decomposition $w = (s_1 \cdots s_{m-1})(s_m)$ yields an endpoint preserving homotopy in $P_3(\Gamma (G,S_k))$ from $w$ to the concatenation of an edge corresponding to an element in $S^{m-1}$ and one in $S$. Proceeding inductively yields the desired homotopy. In the picture $m = 6$.
  • Figure 4: The portion $[v_{i_1}, v_{i_1+1}, \ldots , v_{i_2-1}, v_{i_2}]$ can be replaced by $[a_0, \ldots, a_m]$. In the picture the shadowed region represents $C_j$, the solid line represents $T$, $i_1 = 4$, and $i_2=9$.
  • Figure 5: Each edge of $\Gamma$ is sent via $\psi$ to either a point or an interval of length $\leq 1/4$, so $\psi$ extends to a map $\Psi : P_3(\Gamma) \to \mathbb{R}/\mathbb{Z}$.

Theorems & Definitions (52)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Benjamini--Finucane--Tessera
  • Proposition \oldthetheorem: Švarc--Milnor Lemma
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 42 more