S. M. Gersten Abstract. Homological Dehn functions over R and over Z are introduced to measure minimal fillings of integral 1-cycles by (real or integral) 2-chains in the Cayley 2complex of a finitely presented group. If the group G is the fundamental group of a finite graph of finitely presented vertex-groups Hv and finitely generated edge-groups, then there is a formula for an isoperimetric f...

If K = Goφ Z where φ is a tame automorphism of the 1-relator group G, then the combinatorial area of loops in a Cayley graph of G is undistorted in a Cayley graph of K. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental gr...

It is a difficult problem in general to decide whether a Cayley graph Cay(G; S) is connected where G is an arbitrary finite group and S a subset of G. For example, testing primitivity of an element in a finite field is a special case of this problem but notoriously hard. In this paper, it is shown that if a Cayley graph Cay(G; S) is known to be connected then its fault tolerance can be determin...

In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph G(10, 2), occurring as the...

We examine a number of countable homogeneous relational structures with the aim of deciding which countable groups can act regularly on them. Since a group X acts regularly on a graph G if and only if G is a Cayley graph for X, we will extend the terminology and say that M is a Cayley object for X if X acts regularly on M. We consider, among other things, graphs, hypergraphs, metric spaces and ...

Given a group or semi-group Γ generated (in the sense of semi-groups) by some subset S ⊂ Γ, the Cayley graph of Γ with respect to S is the oriented graph having vertices γ ∈ Γ and oriented edges (γ, γs)s∈S . In this paper, we consider generating sets S which are not necessarily finite thus yielding oriented Cayley graphs which are perhaps not locally finite. The above situation gives rise to a ...

The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomo...

Integral sets of finite groups are discussed and related to the integral Cayley graphs. The Boolean algebra of integral sets are determined for dihedral group and finite abelian groups. We characterize the finite abelian groups as those finite groups where the Boolean algebra generated by integral sets equals the Boolean algebra generated by its subgroups.

We prove several complexity and decidability results for automatic monoids: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the firstorder theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Mor...