Page 338 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 338
SYMMETRY OF GRAPHS, MAPS AND POLYTOPES (MS-9)
Geometry and Combinatorics of Semiregular Polytopes
Egon Schulte, e.schulte@northeastern.edu
Northeastern University, United States
Traditionally, a polyhedron or polytope is semiregular if its facets are regular and its symme-
try group is transitive on vertices. We briefly review the semiregular convex polytopes, and
then discuss semiregular abstract polytopes, which have abstract regular facets, still with com-
binatorial automorphism group transitive on vertices. Our focus is on alternating semiregular
polytopes, with two kinds of regular facets occurring in an alternating fashion. The cuboctahe-
dron is a familiar example in rank 3. We then describe recent progress on the assembly problem
for alternating semiregular polytopes: which pairs of regular n-polytopes can occur as facets of
a semiregular (n+1)-polytope? If time permits, we brief discuss semiregularity in the context of
skeletal polyhedra in 3-space. Most work is joint with Barry Monson.
Regular and ‘half-regular’ maps of negative prime Euler characteristic
Jozef Širánˇ, jozef.siran@stuba.sk
Slovak University of Technology, Slovakia
In 2005 A. Breda, R. Nedela and the presenter classified the (fully) regular maps on surfaces
with negative prime Euler characteristic; this was the first such classification for an infinite
family of surfaces. Extending a 2005 result of M. Belolipetsky and G. Jones, in a 2010 paper by
M. Conder, T. Tucker and the presenter a corresponding orientable version of the classification
was given for orientably-regular maps of genus p + 1 for any prime p.
Algebraically, fully regular maps of valency k and face length correspond to normal quo-
tients of the full (2, k, )-triangle groups presented in the form ∆(2, k, ) = r0, r1, r2; r02, r12, r22,
(r0r2)2, (r2r1)k, (r1r0) . Orientably-regular maps then arise from normal quotients of the
orientation-preserving subgroup r0r1, r1r2 of index two in ∆(2, k, ). Depending on the parity
of k and , however, the group ∆(2, k, ) may contain up to 7 subgroups of index two, giving
rise to further families of ‘half-regular’ maps in addition to the orientably-regular ones.
The above classification results have generated interest in a similar investigation of the re-
maining families of ‘half-regular’ maps. The first family studied from this point of view appears
to be the one of bi-rotary maps which arise from the index-two subgroup r0, r1r2 of ∆(2, k, )
for even. A classification of bi-rotary maps of negative prime Euler characteristic was com-
pleted in 2019 by A. Breda, D. Catalano and the presenter. Recently, another such family of
maps, called edge-biregular and generated by the subgroup r0, r2, r1r0r1, r1r2r1 of index two
in ∆(2, k, ) for both k, even, have been investigated in detail by O. Reade (2021). Moreover,
in a joint 2021 paper by O. Reade and the presenter we have classified edge-biregular maps on
surfaces of negative prime Euler characteristic.
In the talk we will review the previous classification results and present details on the new
ones for edge-biregular maps.
Acknowledgment: This research was supported by the APVV Research Grants 17-0428 and
19-0308, and by the VEGA Research Grants 1/0238/19 and 1/0206/20.
336
Geometry and Combinatorics of Semiregular Polytopes
Egon Schulte, e.schulte@northeastern.edu
Northeastern University, United States
Traditionally, a polyhedron or polytope is semiregular if its facets are regular and its symme-
try group is transitive on vertices. We briefly review the semiregular convex polytopes, and
then discuss semiregular abstract polytopes, which have abstract regular facets, still with com-
binatorial automorphism group transitive on vertices. Our focus is on alternating semiregular
polytopes, with two kinds of regular facets occurring in an alternating fashion. The cuboctahe-
dron is a familiar example in rank 3. We then describe recent progress on the assembly problem
for alternating semiregular polytopes: which pairs of regular n-polytopes can occur as facets of
a semiregular (n+1)-polytope? If time permits, we brief discuss semiregularity in the context of
skeletal polyhedra in 3-space. Most work is joint with Barry Monson.
Regular and ‘half-regular’ maps of negative prime Euler characteristic
Jozef Širánˇ, jozef.siran@stuba.sk
Slovak University of Technology, Slovakia
In 2005 A. Breda, R. Nedela and the presenter classified the (fully) regular maps on surfaces
with negative prime Euler characteristic; this was the first such classification for an infinite
family of surfaces. Extending a 2005 result of M. Belolipetsky and G. Jones, in a 2010 paper by
M. Conder, T. Tucker and the presenter a corresponding orientable version of the classification
was given for orientably-regular maps of genus p + 1 for any prime p.
Algebraically, fully regular maps of valency k and face length correspond to normal quo-
tients of the full (2, k, )-triangle groups presented in the form ∆(2, k, ) = r0, r1, r2; r02, r12, r22,
(r0r2)2, (r2r1)k, (r1r0) . Orientably-regular maps then arise from normal quotients of the
orientation-preserving subgroup r0r1, r1r2 of index two in ∆(2, k, ). Depending on the parity
of k and , however, the group ∆(2, k, ) may contain up to 7 subgroups of index two, giving
rise to further families of ‘half-regular’ maps in addition to the orientably-regular ones.
The above classification results have generated interest in a similar investigation of the re-
maining families of ‘half-regular’ maps. The first family studied from this point of view appears
to be the one of bi-rotary maps which arise from the index-two subgroup r0, r1r2 of ∆(2, k, )
for even. A classification of bi-rotary maps of negative prime Euler characteristic was com-
pleted in 2019 by A. Breda, D. Catalano and the presenter. Recently, another such family of
maps, called edge-biregular and generated by the subgroup r0, r2, r1r0r1, r1r2r1 of index two
in ∆(2, k, ) for both k, even, have been investigated in detail by O. Reade (2021). Moreover,
in a joint 2021 paper by O. Reade and the presenter we have classified edge-biregular maps on
surfaces of negative prime Euler characteristic.
In the talk we will review the previous classification results and present details on the new
ones for edge-biregular maps.
Acknowledgment: This research was supported by the APVV Research Grants 17-0428 and
19-0308, and by the VEGA Research Grants 1/0238/19 and 1/0206/20.
336
