Grid diagrams for higher-dimensional Thompson's groups

Abstract

We describe standard forms for elements of the higher-dimensional Thompson groups $nV$ arising from gridding subdivision processes. These processes lead to standard normal form descriptions for elements in these groups, and sizes of these standard forms estimate the word length with respect to finite generating sets. These gridded forms lead to standard algebraic descriptions as well, with respect to the both infinite and finite generating sets for these groups.

Figures (6)

• Figure 1: A grid square, which can be regarded as a product subdivision of two dyadic subdivisions of the horizontal and vertical directions.
• Figure 2: Two representations of the same element $w$ of $2V$. The top representative is not gridded, but is refined to a grid tree-pair diagram in the bottom representation, where the source square is a product of two dyadic subdivisions of the interval.
• Figure 3: The source square for the example element $w$ and the two trees that define it. The tree $T_h$ is represented on the right of the square because left leaves represent top halves in subdivisions. At the bottom, a tree representing this subdivision where copies of $T_h$ hang from the leaves of $T_v$. An analogous tree representative for the same grid subdivision could be constructed with copies of $T_v$ hanging from the leaves of $T_h$, with the same dyadic blocks occuring, albeit in a different order.
• Figure 4: The process of transforming a diagram of our element $w$ into a grid tree-pair diagram. The top tree-pair corresponds to the element $w$ depicted in Figure \ref{['griddiagram']}. The process consists of adding vertical-type carets (of triangular type, in dotted lines) needed to apply the defining relation for $2V$ and bring vertical carets to the top to form the tree $T_v$. Finally, some horizontal carets (of square type, again dotted) have been added to have full copies of $T_h$ hanging from each leaf of $T_v$.
• Figure 5: Example trees $T_v$ and and $T_h$ used to construct the positive part of the algebraic normal form for an element $v$ as the vertical and horizontal subdivisions in the product, respectively.

Theorems & Definitions (11)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 3.1
• Theorem 3.2
• Theorem 4.1
• Example 5.1
• Theorem 5.2