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On the biautomaticity of CAT(0) triangle-square groups

Mateusz Kandybo

TL;DR

This work investigates biautomaticity for groups acting on locally CAT(0) triangle-square complexes, testing the applicability of Gersten-Short geodesics as a tool for proving biautomaticity. It constructs two compact complexes, $X_1$ and $X_2$, whose universal covers contain radial and infinitely many thoroughly crumpled flats, thereby disproving the conjectures of Levitt–McCammond in general. It then proves $π_1(X_1)$ is biautomatic by passing to a systolic model $X_1'$, while the biautomatic status of $π_1(X_2)$ is resolved affirmatively within the paper, albeit with related questions remaining open. The results clarify limitations of Gersten-Short geodesics for proving biautomaticity in this setting and motivate refined criteria under which biautomaticity may still be established.

Abstract

Following the research from the paper "Triangles, squares and geodesics" (arXiv:0910.5688) of Rena Levitt and Jon McCammond we investigate the properties of groups acting on CAT(0) triangle-square complexes, focusing mostly on biautomaticity of such groups. In particular we show two examples of nonpositively curved triangle-square complexes $X_1$ and $X_2$, such that their universal covers violate conjectures given in the aforementioned paper. This shows that the Gersten-Short geodesics cannot be used as a way of proving biautomaticity of groups acting on such complexes. Lastly we give a proof of biautomaticity of $π_1(X_1)$, however the biautomaticity of $π_1(X_2)$ remains unknown.

On the biautomaticity of CAT(0) triangle-square groups

TL;DR

This work investigates biautomaticity for groups acting on locally CAT(0) triangle-square complexes, testing the applicability of Gersten-Short geodesics as a tool for proving biautomaticity. It constructs two compact complexes, and , whose universal covers contain radial and infinitely many thoroughly crumpled flats, thereby disproving the conjectures of Levitt–McCammond in general. It then proves is biautomatic by passing to a systolic model , while the biautomatic status of is resolved affirmatively within the paper, albeit with related questions remaining open. The results clarify limitations of Gersten-Short geodesics for proving biautomaticity in this setting and motivate refined criteria under which biautomaticity may still be established.

Abstract

Following the research from the paper "Triangles, squares and geodesics" (arXiv:0910.5688) of Rena Levitt and Jon McCammond we investigate the properties of groups acting on CAT(0) triangle-square complexes, focusing mostly on biautomaticity of such groups. In particular we show two examples of nonpositively curved triangle-square complexes and , such that their universal covers violate conjectures given in the aforementioned paper. This shows that the Gersten-Short geodesics cannot be used as a way of proving biautomaticity of groups acting on such complexes. Lastly we give a proof of biautomaticity of , however the biautomaticity of remains unknown.

Paper Structure

This paper contains 8 sections, 14 theorems, 20 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of Complexes
  4. Biautomaticity
  5. Flats in triangle-square complexes
  6. First Example
  7. Second example
  8. Final remarks

Key Result

Theorem 2 .14

Let $X$ be a polyhedral complex in which each cell is at most $2$-dimensional and with finitely many isometry types of cells. $X$ is locally CAT(0) if and only if for each vertex $v\in X$ every injective loop in $\mathop{\mathrm{Lk}}\nolimits(v, X)$ has a length at least $2\pi$.

Figures (13)

  • Figure 1: Left: a triangle-square complex, right: a link of the unique vertex in such complex
  • Figure 2: A part of a radial flat $F$.
  • Figure 3: Points $u_1, u_2, v_1, v_2$ and Gersten-Short geodesics inside flat $F$.
  • Figure 4: A part of a thoroughly crumpled flat $F_3$.
  • Figure 5: The polyhedral complex $X_1$
  • ...and 8 more figures

Theorems & Definitions (56)

  • Definition 2 .1: polyhedral cell
  • Definition 2 .2: polyhedral complex
  • Definition 2 .3: $k$-skeleton
  • Remark 2 .4: covers
  • Example 2 .5: cubical complex
  • Example 2 .6: simplicial complex
  • Example 2 .7: triangle-square complex
  • Example 2 .8: Eisenstein plane
  • Remark 2 .9: constructing complexes
  • Definition 2 .10: length of a curve
  • ...and 46 more