Regular Polytopes With Solvable Automorphism Groups

The starting point of the thesis is the following problem proposed by Schulte and Weiss in[Problems on polytopes,their groups,and realizations,Periodica Math.Hun-garica 53(2006)231-255][1]:Characterize the groups of order 2ⁿor 2ⁿp,with n a positive integer and p an odd prime,which are automorphism groups of regular poly-topes.In this thesis,we aim to solve Schulte and Weiss's problem,and our results on regular polytopes are further used to investigate regualr hypertopes and regular maps of order 2ⁿ.This thesis is origanized as follows.Chapter 1,Introduction:We introduce the history and the background of regular polytopes,regular hypertopes and regular maps,as well as our results in this thesis.Chapter 2,Preliminaries:We introduce some basic definitions and results regard-ing regular polytope,regular hypertope,regular map and finite group.Chapter 3,Regular polytopes:We focus on Schulte and Weiss's open problem.For any positive integers n,k₁,k₂,···,k_d-1such that n?10,k₁,k₂,···,k_d-1?2 and n-1?k₁+k₂+···+k_d-1,we first construct a regular polytope of Schl¨afli type{2^k1,2^k2,···,2^kd-1}and the order of its automorphism group is 2ⁿ.Following the results of Conder in[2],we obtain the sufficient and necessary conditions for the existence of regular polytopes of order 2ⁿ.Furthermore,we classify regular polytopes that have automorphism groups of order 2ⁿand Schl¨afli types{4,2^n-3},{4,2^n-4},{4,2^n-5},{8,2^n-4}and{8,2^n-5}.Finaly,we prove that if the type of a regular 3-polytope of order 2ⁿp is{k₁,k₂},then p|k₁or p|k₂.Up to duality,there are two Schl¨afli types:Type(1)with k₁=2^sp,k₂=2^tand Type(2)with k₁=2^sp,k₂=2^tp.We then show that there exists a regular 3-polytope of order 2ⁿp with Type(1)when s?2,t?2 and n?s+t+1.For Type(2),there exists a regular 3-polytope of order 2ⁿ·3 with type{6,6}.Chapter 4,Regular hypertopes:For any positive integers n?10,s?2,t?2,l?1and n?s+t+l,we construct an infinite family of regular 3-hypertopes of type(2^s,2^t,2^l),whose automorphism group has order 2ⁿ.This answers an open problem proposed by Conder in[3].Chapter 5,Regular maps:We consider regular maps of order 2ⁿfor n?12.(Con-der classified all regular maps of order 2ⁿfor n?11).Let 2?s,t?n-2 and s?t.We show that if s+t?n or s+t>n with s=t,there exists a regular map of order 2ⁿwith type{2^s,2^t},and we conjecture that there is no regular map of order 2ⁿwith type{2^s,2^t}if s+t>n and s<t.We confirm the conjecture for t=n-2 and n-3.Finaly,regular maps of order 2ⁿwith types{2^n-2,2^n-2}and{2^n-3,2^n-3}are classified.Chapter 6,Conclusion:We summarize the main results and propose some open problems for further research.