Asymptotically rigid mapping class groups of infinite graphs

TL;DR

This paper introduces asymptotically rigid mapping class groups for infinite graphs, explores their properties based on the graph's ends, and presents a new class of groups distinct from known Houghton groups.

Contribution

It defines and analyzes a new class of groups derived from infinite graphs, providing explicit presentations and establishing their novelty compared to existing Houghton groups.

Findings

Finiteness properties depend on the number of ends of the graph.

Constructed explicit presentations for pure graph Houghton groups.

Showed these groups are not commensurable with other known Houghton groups.

Abstract

We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the graph has finitely many ends, we construct an explicit presentation for the so-called pure graph Houghton group and investigate several of its algebraic and geometric properties. Additionally, we show that the graph Houghton groups are not commensurable with other known Houghton-type groups, namely the classical, surface, braided, handlebody, and doubled handlebody Houghton groups, demonstrating that this graph-based construction defines a genuinely new class of groups.