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1. CMB Online first
Dihedral Groups of order $2p$ of Automorphisms of Compact Riemann Surfaces of Genus $p-1$ In this paper we prove that there is only one conjugacy class of
dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral
symplectic group can be realized by an analytic automorphism
group
of compact connected Riemann surfaces of genus $p-1$. A pair of
representative generators of the realizable class is also given.
Keywords:dihedral group, automorphism group, Riemann surface, integral symplectic matrix, fundamental domain Categories:20H25, 57M60 |
2. CMB 2009 (vol 52 pp. 366)
A Class of Cellulated Spheres with Non-Polytopal Symmetries We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d-1}$.
We prove that these cellulations cannot be polytopal with maximal
combinatorial symmetry. Such non-realizability phenomenon was first
described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and,
to the knowledge of the author, until now there have not been any
known examples in higher dimensions. As a starting point for the
construction, we introduce a new class of (Wythoffian) uniform
polytopes, which we call duplexes. In proving our main result,
we use some tools that we developed earlier while studying perfect
polytopes. In particular, we prove perfectness of the duplexes;
furthermore, we prove and make use of the perfectness of another
new class of polytopes which we obtain by a variant of the so-called
$E$-construction introduced by Eppstein, Kuperberg and Ziegler.
Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope Categories:52B11, 52B15, 52B70 |
3. CMB 2007 (vol 50 pp. 206)
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ |
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional
$CW$-complex of the homotopy type of an $n$-sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with
free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as
well.
Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space form Categories:55M35, 55P15, 20E22, 20F28, 57S17 |