Short, Highly Imprimitive Words Yield Hyperbolic One-Relator Groups

Figures & data

Fig. 1 Graphs for $|w{|}_a=2$ in the proof of Proposition 1.2.

Fig. 2 Graphs for $|w{|}_a=3$ in the proof of Proposition 1.2.

Fig. 3 A graph for $|w{|}_a=4$ in the proof of Proposition 1.2.

Fig. 4 Number of connected components of the SLPCI${}^{\pm }$ graph by size in rank 3 at length 15.

Table 1 The number of length L $\text{Aut}({\mathbb{F}}_r)$ equivalence classes of cyclic subgroup not contained in a proper free factor of ${\mathbb{F}}_r$, for $L\le 16$.

L	${\mathbb{F}}_1$	${\mathbb{F}}_2$	${\mathbb{F}}_3$	${\mathbb{F}}_4$
1	1	0	0	0
2	1	0	0	0
3	1	0	0	0
4	1	2	0	0
5	1	3	0	0
6	1	8	1	0
7	1	12	5	0
8	1	34	18	2
9	1	71	98	5
10	1	217	522	35
11	1	515	3,124	315
12	1	1,423	16,866	7,106
13	1	3,834	96,086	93,460
14	1	11,816	582,844	1,124,764
15	1	33,321	3,481,458	11,679,597
16	1	95,440	19,514,686	109,264,221

Table 2 The number of length L $\text{Aut}({\mathbb{F}}_r)$ equivalence classes of cyclic subgroup not contained in a proper free factor, by rank and imprimitivity rank, for $14\le L\le 16$.

L = 14	irank	${\mathbb{F}}_1$	${\mathbb{F}}_2$	${\mathbb{F}}_3$	${\mathbb{F}}_4$
	1	1	12	5	0
	2	0	11804	364	6
	3	0	0	582475	321
	4	0	0	0	1124437
L = 15
	1	1	3	0	0
	2	0	33318	258	7
	3	0	0	3481200	1055
	4	0	0	0	11678535
L = 16
	1	1	34	18	2
	2	0	95406	2765	111
	3	0	0	19511903	11023
	4	0	0	0	109253085

Fig. 5 The number of $\text{Aut}({\mathbb{F}}_4)$ equivalence classes of cyclic subgroups of length at most 16 by stable commutator length and imprimitivity rank.