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Register a new userOn planar Cayley graphs and Kleinian groups
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Agelos Georgakopoulos
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Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X subseteq mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y subseteq mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere.
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Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order $9^iv^4$ for any odd positive integer $v>1$ and any $i=0,1$. His result covers all orders of nonelementary abelian groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhills construction. In particular, we show that for a positive integer $m$ and elementary abelian groups $G_i$, $i=1,2,ldots,s$, of order $q_i^4$ such that $2m,|,q_i+1$, there exists a decomposition of the complete graph on the abelian group $G=G_1times G_2times cdotstimes G_s$ by strongly regular Cayley graphs with negative Latin square type parameters $(u^2,c(u+1),- u+c^2+3 c,c^2+ c)$, where $u=q_1^2q_2^2cdots q_s^2$ and $c=(u-1)/m$. Such strongly regular decompositions were previously known only when $m=2$ or $G$ is a $p$-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.
Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X times K_2$ (also called the canonical double cover of $X$) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $K_2$). This means that $X$ is stable. The proof is short and elementary. The theory of direct products implies that $K_2$ can be replaced with members of a much more general family of connected graphs.
In this paper, we construct Kleinian groups $Gamma<mathrm{Isom}(mathbb{H}^{2n})$ from the direct product of $n$ copies of the rank 2 free group $F_2$ via strict hyperbolization. We give a description of the limit set and its topological dimension. Such construction can be generalized to other right-angled Artin groups.
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $dleq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $dgeq4$. Even for this problem, it was only solved for the cases when either $dleq 5$ or $d=7$ and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when $dgeq 11$ is a prime and the vertex stabilizer is solvable.
We show that if G is a finite group whose commutator subgroup [G,G] has order 2p, where p is an odd prime, then every connected Cayley graph on G has a hamiltonian cycle.
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