Compressed decision problems in hyperbolic groups

TL;DR

It is proved that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups.

Abstract

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group $G$, the compressed knapsack problem in $G$ is ${\mathsf{NP}}$-complete.

Figures (9)

• Figure 3.1: A geodesic triangle in a hyperbolic metric space. Note how the three sides "bow in" to a common centre. Dotted lines represent paths of length at most $\delta$ between corresponding points.
• Figure 3.2: Splitting a geodesic quadrilateral according to Lemma \ref{['lem-quad']}.
• Figure 5.1: The evaluation of $B\langle a, b \rangle\langle a', b' \rangle$ agrees with the evaluation of $x \cdot B[i:j]\langle a", b" \rangle \cdot y$. Here we are assuming that $\mathsf{eval}(B) = u = u'u"u"'$ and that $\mathsf{eval}(B[i:j]\langle a", b" \rangle) = w"$.
• Figure 5.2: Case \ref{['sub-both-long']} from the proof of Lemma \ref{['lem-TSLP-SLP']}. Dashed lines represent words that are given by straight-line programs.
• Figure 5.3: Case \ref{['sub-one-long']} from the proof of Lemma \ref{['lem-TSLP-SLP']}. Again, dashed lines represent words that are given by straight-line programs.

Theorems & Definitions (54)

• Corollary 1.1
• Lemma 2.1
• Definition 2.2
• Remark
• Remark 3.1
• Remark
• Lemma 3.2: ECHLPT
• Lemma 3.3: ECHLPT
• Remark
• Lemma 3.4