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This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for `small' genus), by showing that every non-perfect hyperbolic ordinary triangle group $\Delta^+(p,q,r) = \langle\, x,y \ | \ x^p = y^q = (xy)^r = 1 \,\rangle$ has a smooth finite soluble quotient of derived length $c$ for some $c \le 3$, and infinitely many such quotients of derived length $d$ for every $d > c$.

We construct two infinite families of locally toroidal chiral polytopes of type $\{4,4,4\}$, with $1024m^2$ and $2048m^2$ automorphisms for every positive integer $m$, respectively. The automorphism groups of these polytopes are solvable groups, and when $m$ is a power of $2$, they provide examples with automorphism groups of order $2^n$ where $n$ can be any integer greater than $9$. (On the other hand, no chiral polytopes of type $[4,4,4]$ exist for $n \leq 9$.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, {\em Periodica Math.\ Hungarica\} 53 (2006), 231-255]., Comment: 10pages

A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than $2$. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as $B(k,m,n)$, defined for every triple $(k,m,n)$ of integers greater than $1$ with $2^m \equiv 1 \mod n$. In each case, $B(k,m,n)$ is a $2k$-valent vertex- and edge-transitive graph of order $mn^{k-1}$, and Bouwer showed that $B(k,6,9)$ is half-arc-transitive for all $k > 1$. For almost 45 years the question of exactly which of Bouwer's graphs are half-arc-transitive and which are arc-transitive has remained open, despite many attempts to answer it. In this paper, we use a cycle-counting argument to prove that almost all of the graphs constructed by Bouwer are half-arc-transitive. In fact, we prove that $B(k,m,n)$ is arc-transitive only when $n = 3$, or $(k,n) = (2,5)$, % and $m$ is a multiple of $4$, or $(k,m,n) = (2,3,7)$ or $(2,6,7)$ or $(2,6,21)$. In particular, $B(k,m,n)$ is half-arc-transitive whenever $m > 6$ and $n > 5$. This gives an easy way to prove that there are infinitely many half-arc-transitive graphs of each even valency $2k > 2$., Comment: 16 pages, 1 figure

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer $k$, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is $kp$ for some prime $p$, and that if $d\ge 4$, then there exist only finitely many connected $d$-valent 2-arc-transitive graphs whose order is $kp$ or $kp^2$ for some prime $p$. We also prove that there are infinitely many (even) values of $k$ for which there are only finitely many connected 3-valent symmetric graphs of order $kp$ where $p$ is prime.

Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If $\Gamma_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $\Gamma_2$ is a lattice in any connected Lie group, then $\C(\Gamma_1) = \C(\Gamma_2)$ implies that $\Gamma_1$ is isomorphic to $\Gamma_2$. If $F$ is a free group and $\Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(\Gamma)$ implies that $F\cong\Gamma$. If $\Gamma_1<{\rm{PSL}}(2,\Bbb C)$ and $\Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $\C(\Gamma_1)= \C(\Gamma_2)$ implies that $G \cong {\rm{PSL}}(2,\Bbb C)$ and that $\Gamma_2$ belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them., Comment: Minor edits. Version accepted by Israel J Math

The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and \v{S}koviera and (independently) Lovre\v{c}i\v{c}-Sara\v{z}in characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of GI-graphs. For any positive integer n and any sequence j_0,j_1,....,j_{t-1} of integers mod n, the GI-graph GI(n;j_0,j_1,....,j_{t-1}) is a (t+1)-valent graph on the vertex set Z_t x Z_n, with edges of two kinds: - an edge from (s,v) to (s',v), for all distinct s,s' in Z_t and all v in Z_n, - edges from (s,v) to (s,v+j_s) and (s,v-j_s), for all s in Z_t and v in Z_n. By classifying different kinds of automorphisms, we describe the automorphism group of each GI-graph, and determine which GI-graphs are vertex-transitive and which are Cayley graphs. A GI-graph can be edge-transitive only when t < 4 or equivalently, for valence at most 4. We present a unit-distance drawing of a remarkable GI(7;1,2,3).