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A non-plain virtually free group with a strictly 2-geodetic Cayley graph

  • André Carvalho
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Abstract

We construct an explicit infinite virtually free group G=C4∗C2D3 with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of G has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a k-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that G is freely indecomposable and hence non-plain. Consequently, Shapiro's conjecture that every geodetic group is plain and the conjecture that every hyperbolic k-geodetic group is geodetic cannot both hold.

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Cite

BibTeXRIS
BibTeX
@misc{carvalho2026nonplain,
  title = {{A non-plain virtually free group with a strictly 2-geodetic Cayley graph}},
  author = {André Carvalho},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04491},
  eprint = {2610.04491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04491},
}
RIS
TY  - UNPB
TI  - A non-plain virtually free group with a strictly 2-geodetic Cayley graph
AU  - André Carvalho
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04491
UR  - https://arxiv.org/abs/2610.04491
AB  - We construct an explicit infinite virtually free group G=C_4∗_C_2D_3 with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of G has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a k-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that G is freely indecomposable and hence non-plain. Consequently, Shapiro's conjecture that every geodetic group is plain and the conjecture that every hyperbolic k-geodetic group is geodetic cannot both hold.
ER  -